In geometric terms, the figure has is symmetric on rotation (in 90 degree or π/4 radian increments) and reflection (across axes that go from vertex to vertex, or from midpoint to midpoint of the edges).

In group theoretic terms, the figure has Z mod 4 symmetry for rotations, where Z refers to integers, and mod 4 means evenly divisible by 4. Without reflections, the figure can be represented by a dihedral group, again mod 4 with the same meaning.

Further symmetries can be found if this 2-D figure is assumed to exist in higher-dimensional space. If you image it on the x and y axes of a 3-d space, then it could also be rotated or reflected with respect to the z axis. Further extensions into higher dimensions are of course possible.

A good discussion of the symmetries of the platonic solids (this square-like shape is a 2-D projection of a cube) can be found in Artin or similar materials on abstract algebra.

To be fair to your university graduate father, this kind of question mostly comes up in abstract algebra or group theory, typically taught to first-year graduate students in mathematics, though often included in an undergraduate math curriculum.