I think this sequence represents the events of the series as Anthy remembers them: her and Utena coming together and yet being constantly separated.
We see her and Utena together, doing that almost-kiss thing before their hands are torn apart, alternately naked and wearing their dueling outfits. Next we see them in their school clothes, Utena beside a phallic tower and a crowd of boys, Anthy beside a yonic gate and a crowd of girls, their everyday lives apparently separated by gender roles. After that, a pair of cage-like gates part and we see the birdcage-shaped greenhouse in which the two of them stand together, united but perhaps trapped. The lyrics take a bittersweet turn. We’re treated to an unabashedly romantic idyll—something the like of which is never shown onscreen in the series itself—presumably taking place some ordinary day when nothing else was going on. It is interrupted by a transition to the duels.
A montage of duelists. Apocalyptic imagery signals the final duel, revolution. As the arena crumbles, Dios wakes—is this a bittersweet fake-out, with Dios as an empty mirage, or is this a symbol of the awakening of Anthy’s true self?* In another ambiguous moment, we see Anthy and Utena, armor-clad, riding magic horses through the upside-down castle—are they fighting their way towards adulthood and freedom, as the lyrics seem to suggest, or are they trapped in Ohtori’s endless carousel of fairy-tale illusion? The two are torn apart again. Utena falls back and ends up alone.
The end of the opening sequence plays with the idea of an ending in which Anthy vanishes from Utena’s life. Instead, the series shows Anthy constantly, quietly vanishing—denying herself, colluding with Akio—until, ultimately, she is able to imagine Utena waiting for her.
*It’s never made explicit, but I read Dios as largely a projection of Anthy’s own strength and nobility, rather than a lost part of Akio. It seems obvious that Anthy magically imprisoned young Dios’s powers inside herself, and I see no reason to think they were ever his to begin with; perhaps she used witchcraft to make all his good deeds possible. Nor do we see anyone but Anthy behind the Rose Gate. There’s one other hint that Dios is Anthy—when Anthy disguises herself as a boy, her hair looks like Dios’s.