Square is one of those self checking measurements by reversal, but like flat surfaces, you need three squares to agree with each other before you can be sure.
No, you don't need three surfaces. You only need a square and verify it. And after that, your verification instrument is as accurate as you want.
Verifying a square is not so complicated.
But first, standards allow quite a huge (I don't have the numbers here right now) deviation from true squareness. It is almost shocking. So my square standard is G00.
So how do you make a perfect square?
We need a gizmo. It looks like a magnetic stand. Just with a column quite beefy. And a rack along it. The clamp that holds the horizontal bar needs to have a perfect fit and a gear that meshes with the rack. So with a knob, you can drive a DI up/down. And a beefy foot. I have one of those from Mahr. Just paid 20 €-
Next step is, how to verify wether the column is dead at 90°:
Take a square, the precision almost doesn't matter. Put that side to side with the rack-gizmo-thing, so the horizontal legs both point in the same direction.
Crank the rack-gizmo-thing down and mount a DI sticking sideways and having contact with the angles vertical face. Zero. Crank up and note the difference.
Rotate the rack-gizmo 180°, so the legs point one left, the other right. Crank up again and note the difference. Half the difference is how much the gizmo is out of square. Adjust, repeat, until satisfied.
Now you have a rack-gizmo that is dead 90° and you can use that to:
a) adjust your square (by peen hammering)
a) check your CI square.
Maybe I need to make pictures …
Nick