Why do left and right switch when direction is reversed?

Question

If I make a left turn during a trip, it becomes a right turn on my way back. Why is this?

Answer 1

Because turning around to make the trip back corresponds to applying a reflection. A reflection is an orientation inverting operation, so left interchanges with right.

Answer 2

Let $$\gamma:(0,1)\to\Bbb R^2,$$ be a smooth parametrized curve such that $\gamma'(t)\neq 0$ for all $t$ (you don't stop during $(0,1)$). Then, for each $t$ there are two unit vectors $N(t)$ orthogonal to $\gamma'(t)$, only one of which makes $\{\gamma'(t),N(t)\}$ an oriented basis for $\Bbb R^2$. This choice of $N(t)$ gives a smooth vector field along $\gamma$ telling us the left direction. Now, you turn left at $t_0$ if and only if $$\gamma''(t_0)\cdot N(t_0)>0.$$ Taking the reverse curve $$\bar{\gamma}:(0,1)\to\Bbb R^2,\quad\bar{\gamma}(t)=\gamma(1-t)$$ and repeating the process of generating a unit orthogonal oriented vector field $\bar{N}$ for this curve, you will find $\bar{\gamma}(1-t_0)=\gamma(t_0)$ and $$\bar{\gamma}''(1-t_0)\cdot \bar{N}(1-t_0)<0,$$ meaning you now turn right at that time.