Let $\ f:X \rightarrow {\rm I\!R}$ be a function and $X=\{(x,y)\in {\rm I\!R}^2\}$. The explicit expression of this function is unknown, but it can be assumed to be smooth and continuous.
It is know that this function has mirror symmetry (time reversal symmetry in physics jargong), $$ f(x,y)=f(-x,-y), \tag{1} $$ and that the function is periodic (translational symmetry) $$ f(x,y)=f(x\pm2\pi,y)=f(x,y\pm\pi)=f(x\pm2\pi,y\pm\pi). \tag{2} $$ Using the periodicity of the function we can create a closed interval $f:Y\rightarrow {\rm I\!R}$ with $Y=\{(x,y) \in {\rm I\!R}^2\ |\ 0\leq x < 2\pi,\ 0\leq y < \pi\}$. Then by the extreme value theorem a global minima point $(x^*,y^*)$ exists where $f(x^*,y^*)\geq f(x,y)$ for all $(x,y)\in Y$ in this specified closed interval.
Now I'm trying to figure out the subdomain $Y'$ for which the function $f(x,y)$ has a strict global minimum, i.e. a point where $f(x^*,y^*)>f(x,y)$ for all $(x,y)\in Y'$with $x\neq x^*$ and $y\neq y^*$.
Question: Is it possible to figure this out without knowing the explicit expression of the function $f(x,y)$?
My thinking: Because of the symmetries we know that there exists two global minimas $(x^*,y^*)$ and $(2\pi-x^*,\pi-y^*)$ in the domain $Y=\{(x,y) \in {\rm I\!R}^2\ |\ 0\leq x < 2\pi,\ 0\leq y < \pi\}$.
Proof: Assume that the point $(x^*,y^*)$ is a global minimum in the domain $Y$. Begin by using the translational symmetry $$ f(x^*,y^*)=f(x^*-2\pi,y^*-\pi), $$ then make the variable substitution $x'=2\pi-x^*$ and $y'=\pi-y^*$ so that $$ f(x^*,y^*)=f(-x',-y')=f(x',y'). $$ Where we made use of the mirror symmetry. Hence the point $(x',y')$ is also a global minimum in the domain.
A function that satisfies the aforementioned symmetries is for example $$ f(x,y)=\sin x \sin 2y. $$ This function has a strict global minimum over the subdomain $Y'=\{(x,y)\in{\rm I\!R}^2\ |\ -\frac{\pi}{2}\leq x <\frac{\pi}{2},\ -\frac{\pi}{4}\leq y <\frac{\pi}{4}\}$. But I don't know if this applies to every function with these symmetries?
From the conditions you gave we can only determine the maximum such subdomain, and that is a rectangle of dimensions $2π×π,$ with one vertex at the origin, and two edges along the coordinate axes. However, the other two edges are not part of this subdomain.