Let $\alpha:\mathbb{S}^1\to \mathbb{R}^2$ be a $C^2$ curve satisfying $|\dot \alpha|=1$, and define $s= \alpha \times \dot \alpha$, where $\times$ denotes the cross-product of vectors in $\mathbb{R}^3$. (I think of $\alpha, \dot \alpha$ as lying in the plane $z=0$ inside $\mathbb{R}^3$).
Suppose that $s(\theta)$ is non-constant, has no zeroes, and that $ s(\theta+c)=s(\theta) $ holds for every $\theta$, where $c$ is a constant.
Does $\alpha(\theta+c)=\alpha(\theta)$ or $\alpha(\theta+c)=-\alpha(\theta)$ hold for every $\theta$?
(The latter option can happen for instance when $\alpha$ is a non-circular ellipse.)
If we would allow $s=\alpha \times \dot \alpha$ to be constant, then the answer would be negative, in general, e.g. when $\alpha$ is a parametrizing a square, centered around the origin. (Clearly if $s$ is constant, then the condition $s(\theta+c)=s(\theta)$ holds trivially, for every $c$.).
A counterexample would be a (sufficiently nontrivial) curve with rotational symmetry of order $>2$ around the origin. More precisely:
Chose any non-constant periodic $\mathcal C^2$ function $g:\mathbb R\to (0,\infty)$ with period $\frac{2\pi}{n}$ for some $n\ge 3$.
Let $\gamma$ be the curve $\theta\mapsto g(\theta)(\cos\theta,\sin\theta)$.
Let $\beta$ be the curve $\theta\mapsto \frac{2\pi}{\mathop{\mathrm{arclength}} \gamma}\gamma(\theta)$ -- that is, $\gamma$ scaled to have total arc length $2\pi$.
Let $\alpha$ be $\beta$ reparameterized by arc length.
Then $\alpha$ satisfies your assumptions with $c=\frac{2\pi}{n}$, but doesn't satisfy $\alpha(\theta+c)=\pm\alpha(\theta)$.
(Here $s$ will not be constant because it differs between the minimums and maximums of $g$, where in both cases $|s|=|\alpha|$).
I think that with the various non-triviality requirements in the question, all counterexamples will have essentially this form.