Generalization of even-odd parity (symmetry) decomposition of 2-dim functions

Question

For any "well-behaving" real-valued (1-dim) function $f(x)$, the following decomposition that can always be done is basic and at the same time very useful. $$f(x) = f_{ev}(x) + f_{od}(x) \qquad \begin{cases} \text{even:}& f_{ev}(x):= \frac{1}2 \left( f(x) + f(-x) \right) \\ \text{odd:}& f_{od}(x):= \frac{1}2 \left( f(x) - f(-x) \right) \end{cases}$$ It is ubiquitous in analysis for PDE (orthonormal basis), integral equations, and sometimes functional equations.

1. Where can I find a exposition/discussion on a 2-dim analogue (generalization) that is powerful in a similar way? In particular, I assume for 2-dim some geometrical considerations should be prominent.
2. If no such material readily available, what might be some good use or issues with my try below?

The generalization briefly mentioned in the second half of Wolfram Mathworld entry or the section in wikipeida is kind of boring, and in my opinion no where near "complete".

My try to construct something similar to orthogonal basis:

Here the term "even" is generally understood as "symmetric" for the relevant transformation, and "odd" is understood as "anti-symmetric". $$\begin{aligned} f_{Rev}(x,y) &:= \frac16 \left( f(x,y) - f(-x,-y) \right) \qquad \text{"R" for rotation by $\pi$ WRT...}\\ f_{Rod}(x,y) &:= \frac16 \left( f(x,y) - f(-x,-y) \right) \qquad \text{... (with respect to) the origin}\\ f_{Xev}(x,y) &:= \frac16 \left( f(x,y) + f(-x,y) \right) \\ f_{Xod}(x,y) &:= \frac16 \left( f(x,y) - f(-x,y) \right) \qquad \text{reflection WRT $y$-axis} \\ f_{Yev}(x,y) &:= \frac16 \left( f(x,y) + f(x,-y) \right) \\ f_{Yod}(x,y) &:= \frac16 \left( f(x,y) - f(x,-y) \right) \qquad \text{reflection WRT $x$-axis} \\ \end{aligned}$$ These piece are sort of "orthogonal" just like in 1-dim how the sign function (the "core" of odd functions) is orthogonal to the constant function as in $\int 1 \cdot \mathrm{sign}(x) dx = 0$.

The decomposition of "any" 2-dim function then looks like $$f = f_{Rev} + f_{Rod} + f_{Xev} + f_{Xod} + f_{Yev} + f_{Yod}$$ with the behavior the individual components analogous to the 1-dim case: $$\begin{aligned} f_{Rev}(x,y) &= f_{Rev}(-x,-y) \\ f_{Rod}(x,y) &= -f_{Rod}(-x,-y) \\ f_{Xev}(x,y) &= f_{Xev}(-x,y) \\ f_{Xod}(x,y) &= -f_{Xod}(-x,y) \\ f_{Yev}(x,y) &= f_{Yev}(x,-y) \\ f_{Yod}(x,y) &= -f_{Yod}(x,-y) \\ \end{aligned}$$ The above seems okay, but I feel like either something's missing or there are actually already redundancies.

The missing components might are the "prominent geometrical considerations" I mentioned above, which might include rotation by $\pi/2$ and by $3\pi/2$.

The redundancy part is I'm not so sure how to deal with the fact that "rotate by $\pi$ with respect to the origin" is equivalent to the composition "[reflection WRT $x$-axis]$\circ$[reflection WRT $y$-axis]".