We provide a refinement of the Poincaré inequality on the torus ${\mathbb{T}}^d$: there exists a set $\mathcal{B}\subset {\mathbb{T}}^d$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha }>0$ with $${\parallel \mathrm{\nabla }f\parallel }_{L^2({\mathbb{T}}^d)}^{d-1}{\parallel \langle \mathrm{\nabla }f,\alpha \rangle \parallel }_{L^2({\mathbb{T}}^d)}\ge c_{\alpha }{\parallel f\parallel }_{L^2({\mathbb{T}}^d)}^d\text{for all}f\in H^1\left({\mathbb{T}}^d\right)\text{with mean 0.}$$ The derivative $\langle \mathrm{\nabla }f,\alpha \rangle$ does not detect any oscillation in directions orthogonal to $\alpha$, however, for certain $\alpha$ the geodesic flow in direction $\alpha$ is sufficiently mixing to compensate for that defect. On the two-dimensional torus ${\mathbb{T}}^2$ the inequality holds for $\alpha =(1,\sqrt{2})$ but is not true for $\alpha =(1,e)$. Similar results should hold at a great level of generality on very general domains.