Case with VESA mount. How?

OK, I'm thinking of building an ultra-small PC, one with a VESA case that can be mounted on the back of a monitor attaching to the VESA mount on the monitor. Example case (not the one I necessarily want to use, but just one to illustrate what I'm talking about).

Looking at my monitors, my old one has a stand, leaving the VESA mount available. My other, better-quality ASUS monitor has the stand attached to the VESa mount.

So now my question, being dumb. DO I have to plan to only get a monitor where the VESA mount is free or is there some mechanism or trick to mount a small case and use the VESA mount as the monitor mount? I'm, for instance thinking of a wall-mounted monitor or TV acting as a very large photo frame.
 
You'd have to have a monitor where the VESA is free, or possibly a case with a VESA socket on the other side (don't know if such a thing exists).

TVs use a far larger VESA mount, often 400mm, and generally won't work with those kinds of cases.

You may also want to look in to thin-ITX based AIOs.