The Google Cardboard app does this. (Did this? I haven't used the app in a long time.)
It splits your single 360 pano into 2 images, one for each eye, but it doesn't do any stereo depth stuff. It's like being in the middle of a sphere with everything painter on it all the same distance away.
When there's something 3d rendering the scene it can too stereoscopy, but if there's actual cameras trying to create stereo images in 360 - you always end up with the "what does it do when you're looking directly at the other eye's camera?" paradox.
I played a lot with it a few years back, even wrote a library to read the files in C#.