Does for every Continuous function $f:S^1 * S^1$ to $\mathbb{R}^2$ exist a point for which $f(a,b)=f(-a,-b)$?

Question

Does it hold that for every continuous function $f:S^1*S^1$ to $\mathbb{R}^2$ there exist a point for which $f(a,b)=f(-a,-b)$? I think the answer is positive since by Borsuk-Ulam there is such point if the domain is $S^2$, and $S^2$ contains so many $S^1$ as great circles and any two great circles intersect at least on two points...

Answer 1

What about a projection map? If we view each $S^1$ factor as the standard unit circle in $\mathbb{R}^2$, then $\pi:S^1\times S^1\rightarrow S^1\subseteq \mathbb{R}^2$ with $\pi(a,b) = a$ satisfies $\pi(a,b) = \pi(c,d) \iff a=c$.