Badass Army: Militarily one of the most powerful members of the Multiverse Alliance, with its massive Imperial Army, Imperial Navy and its Space Legions. Justified since the Empress based their military on that of the Imperium from 40K. As of first contact they were engaged on a 'Grand Crusade' to reunite all of Ponykind's lost colonies across the stars, a massive military endeavor that has been going on for centuries and spans their entire galaxy in scope.
Their Army and Navy are no slouches, either, despite being composed-of/crewed-by ordinary ponies: Like their counterparts from 40K, their arsenals ranges from gigantic super-heavy tanks bristling with guns able to pulverize entire enemy regiments, all the way to kilometers-sized starships capable of cratering/glassing whole continents from orbit.
Blue-and-Orange Morality: Downplayed, but evidently observable, and explained to be the direct result of the galaxy they live in. The Hegemony's hard-line attitudes on various matters, notably their views on justice and punishment, unnerve many of their allies: They find nothing wrong with using criminals and generally condemned ponies as indentured labour on starships and penal worlds, or punishing those who committed 'tech-heresy' (E.g. making hostile A.I.s for conquest) by turning them into lobotomized, servile cyborg drones called Servitors.
Crippling Overspecialization: They actually struggle to shoot down the nuclear missile Tales!Chrysalis launched due to their ships being prioritized for space battles and It's the Only Way to Be Sure level situations over hitting a comparably tiny, fast moving target at range. It's noted this is a source of embarrassment and shame for them.
The Dreaded: Quickly establishes itself as this to those they encountered, even their allies, between their ruthlessness, military might and hardline attitudes.
The Empire: The Hegemony of Equuskind is a galaxy-spanning interstellar empire ruled by an immortal Empress. Said Empress is none other than an alternate future version of Princess Twilight.
Establishing Character Moment: There was not one but many, many moments over the course of the round-robin which highlighted the well-intended, honourable and determined but also paranoid, aggressive and ruthless nature of the Hegemony, though a few particularly stand out.
When the issue of the Cozy Glows held in captivity across the Alliance following their failed coups was decided, the Hegemony sent one of their Inquisitors, Adamantia, and her entourage of stormtroopers to deal with one of them being held in an alternate Equestria's Tartarus. In a shocking scene which stunned even Lord Tirek, after reading out her charges and hearing her complete and utter lack of regrets, Adamantia immediately sentenced her to death and executed her with a bolter round to the head. The scene served to demonstrate, with utter finality, just how brutal and ruthless the Hegemony can be.
Expy: The Hegemony is a basically a ponified, mostly benevolent expy of the Imperium from WH40K. Makes sense since the Empress outright admits that she based the entire thing on the Imperium, which she visited via interdimensional travel in ages past
The Extremist Was Right: As one of the leading 'Hardliners' among the Multiverse Alliance, they usually advance or advocate the more extreme solutions to problems, as well as being more extreme in views. While they are definitely not always right in their methods (Especially since not all problems are as bad as they ones they experience back in their galaxy), there is a disturbing trend of them being correct in a lot more things and situations than otherwise.
For example, nobody took them seriously when Inquisitorial members of the Alliance's security organizations laid out their suspicion the School Of Friendship student Cozy Glow is up to something nefarious. To everyone's shock and the Hegemony's complete lack of surprise, it turned out they were right. Played for Dramatic Irony with King Onyx of Transneighvania since they remained suspicious of him even after most ruled him out as beyond suspicion. Again, as the readers know, they were right to do so.
Subverted with the situation with Flutterbat: when magically-sensitive members of the Alliance detected her presence in Mainline Fluttershy, Hegemony members immediately moved in to try and 'exorcise' her, seeing her as a threat in the same vein as the hostile Aetherean entities and a potential security risk, which seemingly turned out true when she ratted out many sensitive information about the Alliance to the Vampire Six of Worldline VWE-1010 at First Contact. But it turned out she's actually neither evil nor treacherous, only afraid of the Hegemony and other Hardliners in the Alliance and don't want them to destroy the Vampire/Werewolf-dominated world that she felt kinship with, precisely because they tried to destroy her out of excessive paranoia.
Fusion Fic: Of Equestria with the WH40K universe, albeit with VERY notable spins and differences. Xenos like the Hrud, the Orks (As 'Greenskins') and the Eldar are present and mentioned in narrative, as are the Old Ones. Humanity and Earth are replaced with Ponies and Equus (Plus other races from the setting). It's also a Crossover since that universe's Twilight having traveled and lived in the Imperium of Man from 40K for a time via the Portal Caster
Faster-Than-Light Travel: One of the few members of the Multiverse Alliance, including Space Equestria and Bright Future Equestria, capable of interstellar travel and possess interstellar territories. This involved the use of the Aether Drive, an expy of 40K-verse's Warp Drive which enables FTL travel through the Aetherean, a vast but dangerous parallel dimension which is also the source of magic in their universe.
The Juggernaut: A member of the Multiverse Alliance and is stated as one of their most powerful members in terms of technology, military, economy and population. The fact that they have millions of star systems worth of territory and corresponding resources under their control is repeatedly emphasized. Due to circumstances of their existence they're also one of the most militant and aggressive members of said Alliance.
Lighter and Softer: Actually manages to be this despite being based on the 40K universe, even with the galaxy being a war-torn Crapsack World, due to being set in a time of hope and recovery as the Great Crusade Era was in the backstory of 40K. And it is unlikely that something like the Horus Heresy is ever going happen due to the Empress avoiding the mistakes her namesake committed, as well as due to due the War in Heaven never happening, according to Word of God. This has huge implications for those familiar with 40K lore as it also means the Necrontyr and the C'Tan either never existed or no longer does, and without said conflict the Ruinous Powers of Chaos themselves that would had been spawned from it never existed either.
The Aetherean, this worldline's analogue of the Warp and the source of all magic and FTL, while still scary and dangerous with its unpredictable environment and some hostile Aetherean entities, it is still safer than the Warp, notably due to the the complete absence of daemons and the Ruinous Powers.
Also, unlike in 40K, the Empress got the Old Ones' Webway working again, meaning that at least in this worldline, the 40K Emperor's grand dream of uniting the species across the stars like never before has or will become a reality.
Machine Worship: The Mechanicum, a major organization/component of the Hegemony, is an expy of the Mechanicum from WH40K. Like its 40K counterpart it worships a Machine-God called the Omnissiah, its engineers are red-robed cyborgs called 'Tech-Priests' who venerates the very machines they maintain and use, and has a concept of 'Tech-Heresy', one of which is a hostility to any form of A.I.s, which they term 'Abominable Intelligence'.
Justified in that like in the backstory of 40K, their galaxy had suffered from an AI-rebellion from robots collectively called the 'Mares of Iron', which devastated the previous interstellar pony civilization and led to the significant loss of knowledge and technology.
The Mechanicum are dedicated to recovery of knowledge, holds a monopoly on making and maintaining all Hegemony technology in their galaxy, and are so advanced with what they had so far that they are considered collectively a match for Princess Apple Bloom from Bright Future Equestria, who is a goddess of invention and a supreme Gadgeteer Genius that had individually responsible for inventing/innovating many of her universe's various advances and is classified as a "12th Level Intellect".
Properly Paranoid: Consistently described as paranoid in character, which is explained as inevitable consequences of living in thekind of galaxy they're in (Any of those who are not cautious enough are already dead). And disturbingly, they are more often than not justified in their paranoia, even outside their universe.
Reality Ensues: Their ruthlessness and tendency to want to pragmatically quickly deal with a potential threat before it can become one doesn't endear them to all of the Alliance, who more often than not prefer any possible peaceful resolution to a problem. Often times, the Mane Six actively fear how they will react in any given situation and their representatives butts heads with the more peaceful members of the Alliance on many occasion. In fact, one such instance of this is why Megan was brought into the Alliance to serve as a mediator in the first place. This is largely due to the fact in their home reality, this is a proper and perfectly acceptable reaction to many of the horrors that exist, and thus they clash ideologically with universes were such things are not the case.
Teeth-Clenched Teamwork: As mentioned above, their past history with hostile A.I.s had made the Mechanicum, and the Hegemony as a whole, extremely wary of robots and A.I.s. This had caused considerable tension and conflicts with members of the Multiverse Alliance who either championed sapient robots or are sapient robots. This is expressed during and after the Invasion Arc where they were among those pushing for taking the war to the robot-army using Flim Flam Brothers' world and called for Metal Twilight's destruction even after she have had a HeelFace Turn.
Seen It All: Combined with heavy doses of Conditioned to Accept Horror and Stiff Upper Lip. While they are could find themselves shocked or confused by various things and threats they encounter and fought across the multiverse, they often accept or adapt very quickly to them. Justified since the Hegemony covers myriad of worlds and encountered many strange and/or terrible civilizations, whether Equus-descended or alien, and many of the Hegemony characters are hardened soldiers who had fought for years to centuries across countless battlefields in a war-torn galaxy facing all sorts of threats. As a general rule, if it exists somewhere in the Multiverse, they have likely encountered something similar back home (And likely KILLED it, too).
We Have Reserves: Owing to being a heavily militarized galaxy-spanning empire, the Hegemony possess literally billions upon billions of troops and thousands upon thousands of warships. During the 'Invasion' Arc it was mentioned they contributed nearly 2/3 of all the troops defending Mainline Equus from Flim-Flam Incorporated's invasion forces; enough to help fully defend Equestria AND still allow for deployment
Been There, Shaped History: Been around since the time of Equestria. When the ponies left Equus for the stars, she in fact helped with the development of the Aether Drive that made it possible. Justified since she's an alternate future version of Princess Twilight Sparkle
Thanks to being a Dimensional Traveler herself, she also lived through the entire Age of the Imperium in the WH40K universe itself along with its many events, from the Horus Heresy all the way to the revival of Primarch Guilliman and the Indomitus Crusade ten thousand years later
Dimensional Traveler: Revealed to be one herself, having independently created a Portal Caster and going off exploring the multiverse, first with her friends, then by herself. She had mentioned to have spent life-times in other world-lines. Also explained that's where she got the idea to base the Hegemony on the Imperium of Man from the 40K universe, given the similar circumstances. She also at one point met the Doctor.
Expy: Of the Emperor of Mankind from WH40K. Implied to have known him in person while she was living in the Imperium. She notably does not possess many of the original's flaws nor committed his mistakes, having seen first-hoof what it led to in all its horrifying glory..
Founder of the Kingdom: Founded the Hegemony of Equuskind with her closest companion, servant and friend, Midnight the Sigillite.
Mysterious Past: Who she was and what she had been up to before her appearance is an enigma to the rest of the Hegemony. Owing to tens of thousands of years having passed, even the reveal that she is a far future version of Twilight Sparkle leaves much of what she had gone through and been up to between times of Equestria and the Hegemony unknown and full of mystery. It is shown that at some point she met the Doctor and is on good terms with him.
Physical God: As an immortal Alicorn, she is an actual goddess for all intents and purposes and is seen as such by many of her subjects, even if she prefers not to.
Time Abyss: She is at least 30 millennia which is older than her mainline counterpart. More if accounting for possible variations in flow of time she had spent in other realities. Despite this, she's still pretty much herself.
Who Wantsto Live Forever?: It is noted by the narrative and some of the Mane Six that beneath all her immense power, wisdom and authority, she is a very old and, psychologically, an extremely tired mare who has lost pretty much everything she loved and cherished, including all her friends, family and home from a long time ago. Nevertheless, she pressed on out of her love and duty for her people.
Midnight the Sigillite
The Ageless: Thanks to her proximity to and exposure to the Empress' powers, she's become biologically immortal on top of becoming a powerful Psyker.
Badass Bureaucrat: Help found and run the Hegemony for the Empress as head of the Administratum. but is also a powerful Psyker who can send enemies flying with telekinesis, deflect projectiles with barriers and fry her foes with electrical blasts.
Cool Uncle: Is a Gender-inverted example for the Primares, who fondly referred to her as 'Aunt Midnight'. Unsurprising given she practically helped raised them with the Empress.
Expy: Of Malcador the Sigillite from 40K, effectively sharing the same role and title.
Mysterious Past: Manages to be even more so than the Empress, though there isn't much to begin with, given she was found and rescued by the Empress when she was but a little filly in the wastelands of pre-Hegemony Equus. Her first memory was that of seeing the Empress, with nothing known about her parents or any time before that.