> But what could they be orbiting around?

>> a mutual centre of gravity

The math of gravity is fascinating. Objects can orbit each other, with nothing in the middle.

You think that's good, try this: Objects can even orbit interesting points in space that are not themselves centers of gravity:

https://en.wikipedia.org/wiki/Lagrangian_point

This is linked to orbit around Lagrangian points but it's more surprising: https://en.wikipedia.org/wiki/Horseshoe_orbit

By extension it could be interesting to model the Lagrange-point space topology within this system and probe what object forms might have assembled themselves within the equivalent of L4 and L5 points. Of course, intense mutual solar winds would mitigate against that happening.

Their center of gravity, say about here:

   o O---------------o
         ^

Pluto and Charon do it as well, even Earth and Moon to a small degree.

I've actually heard someone argue that because of this the moon is actually not a moon but our sister planet.

All orbits go around a common center of mass which is not the center of mass of the orbiting bodies separately. The Earth-Moon center of mass is still located within the Earth.

[edit] I forgot to mention that the distinction between a satellite/moon and another co-orbiting body had to do with the location of that barycenter being within one of their bodies, if I recall correctly.