Reversing the first loop changes the sign of the Gauss linking integral

Let $\gamma_1:\mathbb{R}\to\mathbb{R}^4$ be $1$-periodic, let $\gamma_2$ be any loop, and let $c\in\mathbb{R}$. Traversing $\gamma_1$ backwards, $s\mapsto\gamma_1(c-s)$, reverses the sign:

$\mathrm{Gauss}_N(\gamma_1,\gamma_2)$ denotes gaussLinkingIntegral N γ₁ γ₂, the Gauss double integral $\frac1{4\pi}\int_0^1\!\int_0^1\frac{\operatorname{vol}(A'(s),B'(t),A(s)-B(t))}{|A(s)-B(t)|^3}\,dt\,ds$ of the stereographic images $A=\sigma_N\circ\gamma_1$, $B=\sigma_N\circ\gamma_2$ from the pole $N$.

The velocity of the reversed loop is $-\gamma_1'(c-s)$, and the integrand is linear in the first velocity. Substituting $s\mapsto c-s$ and using periodicity gives the claim. No regularity hypothesis is needed. This is $\operatorname{lk}(-K_1,K_2)=-\operatorname{lk}(K_1,K_2)$.