Also in 3D, two reflections make a rotation?

The easiest proof I know for the 2D statement in the summary does not carry over nicely to the 3D statement since rotations in 3D don't necessarily commute (the 2D proof uses this commuting among rotations in the plane around a common point). Before I then try to modify the proof so that it works, I would like to know whether the statement for 3D is even true. Thanks in advance.

I would refer you to Linear and Geometric Algebra by Alan Macdonald, problem 7.3.9:
7.3.9 (1 rotation ##\equiv ## 2 reflections) a. Show that the composition of two reflections in hyperplanes is a rotation, with the angle of rotation twice the angle between the hyperplanes.
b. Show that a rotation is the composition of two reflections in hyperplanes, with the angle between the hyperplanes half the angle of rotation.

I am not sure how well this matches your question, but I think it does.

Super! It answers my question very well. Thank you, FactChecker!