We are pretty confident we understand black holes of a range of sizes, but we still can't say for sure that some fundamental particles are not actually themselves little black holes, or if the whole universe is swarming with little black holes somehow. This idea of white holes isn't discussed much. You wouldn't expect to see one in space, unless you mean by viewing the big bang as a white hole - it is, after all, a region of space that light and matter cannot enter. And it is thought to gravitate; we initially thought expansion would slow from all the matter pulling on itself. A white hole would gravitate, just like a black hole, so you could 'orbit' it, assuming it wasn't spitting out light or matter. And we're kind of orbiting the big bang; only it happened everywhere at once, so that involves a lot of sitting around.

I'm not sure how else one might mean opposite; there is the unruh radiation... like, you could look at the furthest reaches we can see, and think about how we relate to a singularity. There might be some interesting symmetries. It's above my head.

The wave mechanics of gravity are complex; regions of space can focus gravitational waves, or spread them. Space is warped by 'frame-dragging' around a spinning black hole. And we don't quite get either dark matter or dark energy.

In short, the easiest opposite is the time reversal. Black holes are closed-off regions of space; they have properties that seem bizarre to us. Their opposite is either not bizarre, or just bizarre after having been put through a Wick rotation.