Why Moira X is Destiny's antithesis? Because Destiny and Moira X are Foils to each other:
Destiny's powers manifested when she was 13 (according to Mystique in Uncanny X-Men #389, Sage in X-Treme X-Men #1, and Irene herself in Immortal X-Men #3), so did Moira's.
In Rise of the Powers of X, Xavier's secret plan is to travel back in time before Moira's powers activate, that is, when she is 13.
Spurred by her visions, Irene wrote her 13 diaries, putting on the paper visions of futures, about "places she'd never heard of and scores of people yet unborn" (Sage, X-Treme X-Men #1). Also according to Sage, Destiny tried to perceive which of her visions were the true timeline, the most desirable, and if it was possible to combine the two. With the help of Raven Darkhölme, both women tried to steer the course of history to their advantage. That is, Irene is using her foreknowledge of the future.
On the other hand, Moira X had 9 previous lives, and, in her tenth life, allies herself with Xavier before the X-Men even start. With "perfect recollection" of her past lives, she tries to avoid the same mistakes and course correct the world in a way that is most favourable to her. Moira has knowledge of her past lives, and has joined forces with Xavier and Magneto in order to steer history in favour of mutants in her 10th life (before her retcon).
This also explains why Moira did not want any precogs on Krakoa, but especially Irene.
The parallels to each other are further shown when, at the final pages of X-Men #20, Moira is seen reading Destiny's Diaries in her No-Place Zone, although this plot point is not explored further.
One can also see parallels in regards to their relationships: Destiny is Mystique's partner, partner in crime and lover/wife, Mystique being one of the pragmatic leaders of the mutant cause; in turn, Moira was Xavier's love, and, despite breaking things off with him on bad terms, still helped in the advancement of Xavier's vision for the mutant cause (retcon notwithstanding).
