I call the thing to the left an "eigenstate" because if you click all the squares that are initially red, then you will reach the same pattern again. If you click all of the black squares, then you get the same pattern but with inverted colors. If we let black="don't operate" and red="operate", then every pattern can be thought of as a function of space which can also be applied like an operator. Then clicking all of the red squares can be thought of as applying such an operator to itself, F⋅F.

What is this function-operator hybrid? I imagine it's something like the convolution of a 2D function, f, and an operator, A, which will have the property that A is applied [once] where f=1 and is not applied where f=0. We will thus need some sort of exponential analogue of convolution. Naive suggestion:

A⊗f ≡ ΠA(r')^f(r')dr'.

Some other comments:

1. The "eigenstate" E is a case where EE=E, but what's even stranger is that every initially-randomized pattern has a particular corresponding eigenfunction E and clicking all of the blocks of one color will ALWAYS result in E. i.e. For an arbitrary block pattern F, FF=E always.
2. In this example, clicking all of the red squares of E will result in E again, while clicking all of the black squares will result in -E, but this is arbitrary. In fact, this convention is determined when you click all of the squares of a chosen color to get E from F in the first place.
3. The rows of E are always identical to each other barring inversion. The same is true for the columns. This suggests a definition of E in terms of inversion numbers which represent whether a row or column is inverted relative to the first one. This means there should be 2(N-1) such numbers. E is then easily constructed by starting with the solved puzzle and applying the inversions.
4. It's generally impossible to solve the puzzle for the C=2,N=5 case, but in this case there is still a simplest possible state, S. S is different depending on the random initialization of the block colors. For this particular example, it consists of 3 blocks. It might therefore be convenient to use S as a signature of the initial conditions (a signature that is somehow conjugate to the inversion numbers described previously).
5. The fact that the N=4 puzzle can always be solved implies that there is a mapping between any two random game states in terms of the cross-inversion operator associated with clicking. The fact that the N=4 puzzle can always be solved manifests in the fact that it does not have eigenstates like the N=5 puzzle. On the contrary, in the N=4 puzzle, clicking all of the red blocks and then clicking all of the red blocks in the resulting pattern will always solve it.