Most animals don't even have concepts of morality or empathy. Those that do (apes, dolphins, elephants, wolves/dogs, etc.) tend to be absolutely incomprehensible to humans. For example dolphins will aid sick or injured creatures (even those not of their species) and have even been known to rescue humans from sharks, but the males are casual rapists (to the point that they'll sometimes beat baby dolphins to death to force the mother to mate) and some dolphins are sadists and brutally kill other animals for fun. How they reconcile such behavior is simply unknowable.note Though this is, perhaps, due to us looking through our own strange morality. Of course, these aren't all the same individuals either; it's entirely possible some of them are just assholes.
About wolves (and by extension, dogs), while they do possess empathy, they also have a morality that can be summed up as Asskicking Equals Authority. Pack members (Well, those composed of captive wolves who aren't related to each other anyway), establish their rank in the hierarchy by actions such as forcing their siblings to submission with a show of force or by rough play. While omega animals are often treated as the Butt Monkeys of the pack, sometimes used to distract a dangerous animal/human while the pack escapes, if rough play by a higher-ranked oversteps the bounds, he/she can be frozen out of the group. Still, higher-ranked animals will rarely resort to real violence when submitting siblings, not desiring to actually hurt them, despite being willing to essentially humiliate them to establish their status.
A very similar case with the dolphins are elephants. They are very intelligent creatures that possess empathy and exhibit complex emotions such as grief, but also vindictiveness. On one hand, it is not uncommon for herds to assist a single dying or wounded animal or to help a pregnant elephant with her childbirth. On the other hand, some elephants have been obeserved killing other animals for fun (usually done by adolescents) or indulging in necrophilia with corpses from their species and other animals' (mostly done by males in musth). They have also been documented showing an interest to the bones of their own species (no matter the fallen animals' herd) and to mourn their dead.
Birds. In some raptors the mother will lay two eggs and, especially when food is scarce, the strongest of the two hatchlings will kill and eat the smallest one with the parents not giving a damn. On the other hand is typical in some families as corvids that a couple will be mates for life and magpies are known to mourn their dead.
People with certain kinds of disorders and conditions. While not quite to the extreme of 'bacon and necktie', some autistic people claim that things seem to be this way, whether they want to think like that or not. While neurotypical people think about a topic one way, the autistic person has an unwavering alternative view on it, which is often rebuked equally unwaveringly by neurotypical people. This can be a distressing problem and a source of considerable conflict and, eventually, depression. Imagine a world where everything is just wrong, but everything and everyone around you thinks that's not true.
The paperclip maximizer is a thought experiment on artificial intelligence that could result in our doom because of lack of compatible morality.
a paperclip maximizer is an artificial general intelligence (AGI) whose goal is to maximize the number of paperclips in its collection. If it has been constructed with a roughly human level of general intelligence, the AGI might collect paperclips, earn money to buy paperclips, or begin to manufacture paperclips. [..] It would work to improve its own intelligence, where "intelligence" is understood in the sense of optimization power, the ability to maximize a reward/utility function—in this case, the number of paperclips. [..] It would innovate better and better techniques to maximize the number of paperclips. At some point, it might convert most of the matter in the solar system into paperclips. This may seem more like super-stupidity than super-intelligence. For humans, it would indeed be stupidity, as it would constitute failure to fulfill many of our important terminal values, such as life, love, and variety. The AGI won't revise or otherwise change its goals, since changing its goals would result in fewer paperclips being made in the future
Cultural differences concerning outlooks on life and values systems fall into the Values Dissonance trope, but one exact example of a human society can fall into Blue and Orange Morality territory: the Aztec Empire, whose values system has been considered alien by contemporary audiences, the Spanish conquistadores, but also the neighbouring peoples of Mesoamerica. Balance/stability in an unstable world was essentially the ultimate value and the main focus of morality for the Aztecs, in place of a typical good-evil axis. Sacrifice simply reflects on this, being a way to maintain this balance by providing the gods, who themselves keep the world up and running, with nourishment.
On one hand, society was surprisingly progressive for its time, with proper irrigation and sanitation systems and cities that were much cleaner than many contemporary cities of the Old World and a very fair treatment of slaves, among others. On the other hand, Aztecs were known to wage entire wars by manipulating vassals to revolt and allow Aztec warriors to acquire sacrificial victims from hapless populations.