Let $F$ be a $C^1$ function and consider $$G(x) =ax- F(x+a),$$ with $a\in\mathbb{R}^*_+$. I need $G$ to be even. Clearly, if $F$ is odd in the "usual" sense, so it is $G$. But, actually, I need $G$ to be even.
It makes sense to consider on $F$ a sort of "oddness" with respect to $a$ so that $G$ results to be even?
Thank you in advance!
As you evidently know, every function can be written as the sum of an even function and an odd function $$g(x)=\underbrace{\frac{g(x)+g(-x)}{2}}_{\text{even}}+\underbrace{\frac{g(x)-g(-x)}{2}}_{\text{odd}}.$$
These two parts display a very nice symmetry, whether by reflection around the $y$-axis or reflection through the point $(0,0)$.
It would be a remarkably odd (pun intended) fact if this were useful but similar symmetry around any point other than zero had never been attempted.
The short answer to your question is of course this kind of symmetry about other points than zero is useful and makes perfect sense!
In consideration of continuity, differentiability, etc in the calculus you see the expression $f(x+t)-f(x)$ everywhere. If we exploit this odd/even symmetry at the point $x$ we can write $$f(x+t)-f(x)= \frac{f(x+t)-f(x-t)}{2} + \frac{f(x+t)+f(x-t) -2f(x)}{2} .$$
These are called the first symmetric difference and the second symmetric difference and are obvious ways to exploit the odd/even split you are familiar with.
If you think derivatives are important (hint: they are) $$f'(x)=\lim_{t\to 0} \frac{f(x+t)-f(x)}t$$ then you won't be surprised the even and odd versions of the derivative are important: $$ \lim_{t\to 0} \frac{f(x+t)-f(x-t)}{2t} $$ and $$ \lim_{t\to 0} \frac{f(x+t)+f(x-t) -2f(x)}{2t} $$
You will see these kind of expressions in many studies, particularly in Fourier series.
A small sample of an "even" property and an "odd" property. Note that both of these are much better using odd/even parts than just $f(x+t)-f(x)$.
Theorem (Dini) The Fourier series of a $2\pi$-periodic integrable function $f$ converges to $f(x)$ at a point $x$ if $$\int_0^\pi \frac{|f(x+t)+f(x-t) -2f(x)|}{t} <\infty.$$
Theorem (Khintchine) A measurable function $f$ is differentiable at almost every point $x$ at which $$\limsup_{t\to 0} \frac{f(x+t)-f(x-t)}{2t} < \infty.$$