There are two types of trominoes, straight shapes and L-shaped. Suppose a rectangle $R$ admits at least one tiling using trominoes, with an even number of L-trominoes.

**EDIT:** we do not admit ALL orientations, but only those which are constructed by starting in a box, and next box is placed to the left, or below previous box.
Hence, only 2 out of the 4 possible orientations of the L-tromino is allowed. Hence, these are the four trominoes that we are allowed to use:

$$\begin{align}\newcommand{\x}{\large\blacksquare}\newcommand{\o}{\large\phantom{\square}} &\o\o\o\o\o\o\o\o\o\o\x\\[-5pt] &\x\x\o\o\o\o\x\o\o\o\x\\[-5pt] &\x\o\o\o\o\x\x\o\o\o\x\o\o\o\x\x\x \end{align}$$

*Prove that every tiling of $R$ must use an even number of L-trominoes.*

This smells a lot like a classical tiling problem, but unlike the classical cases (dominoes and missing corners), the situation is not to show that a tiling is impossible…. Some nice invariant is perhaps what I am looking for.

I actually know how to prove the above statement, but it requires a lot more machinery than I would like. Moreover, I am interested in a more general question, (regarding representation theory and cylindrical Schur functions, and the proof I know does not generalize to this situation), but I hope that a good proof of the above problem generalizes.

For the interested: The general setting supposes that the rectangle is a torus, and that we use $k$-ribbons, where $k$ is odd. We then want to show that the number of $k$-ribbons which occupies an even number of rows, occur an even number of times. Hence, a proof not relying on the fact that $R$ has boundaries, or heavily uses the fact that each shape has three squares, is of extra interest.

**EDIT2:** 2022-12-09 So, I am in the process of writing up a proof of this (and some other stuff).

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