I've noticed that if I have any curve in polar form $r = f(\theta)$ such that $f$ is periodic with period $2\pi$ and defined for all real numbers, then the curve $r = -f(\theta)$ is a 180° rotation of the first curve around the origin and it's equation is also equal to $r = f(\theta + \pi)$.
Intuitively, negating the argument can be though of as plotting the graph "backwards" in the clockwise direction. It sounds plausible that plotting the graph in such a way is the same as rotating it 180°, but I'm not 100% confident in that.
How to prove that these two forms of the curve ($r = -f(\theta)$ and $r = f(\theta + \pi)$) are indeed equivalent? It seems like it should be trivial, but I'm not sure where to even start.
Let the curve $\gamma$ be the set of points whose polar coordinates satisfy the equation $r = f(\theta)$. That is, the point $P_1 = (r_1,\theta_1)$ is on the curve $\gamma$ if and only if $r_1 = f(\theta_1).$
Now let's rotate the point $P_1$ with coordinates $(r_1,\theta_1)$ by $180^\circ$ clockwise around the origin. This arrives at the point $P_1'$ with coordinates $(r_1',\theta_1') = (r_1,\theta_1 - \pi).$ It's easy to see that $r_1 = f(\theta_1)$ if and only if $r_1' = f(\theta_1' + \pi),$ so a formula of the rotated curve is indeed $r = f(\theta + \pi).$
It should also be clear that when you rotate the point $P_1$ with coordinates $(r_1,\theta_1)$ by $180^\circ$ clockwise around the origin, you end up at a point on the same line through the origin, at the same distance from the origin, but on the opposite side of the origin. That's exactly how we interpret the polar coordinates $(-r_1,\theta_1)$. Therefore $P_1'$ also has coordinates $(r_1'',\theta_1'') = (-r_1,\theta_1)$. And here it's easy to see that $r_1 = f(\theta_1)$ if and only if $r_1'' = -f(\theta_1''),$ so a formula of the rotated curve is indeed $r = -f(\theta).$
Another way to put it is that every point in the plane has an infinite number of sets of polar coordinates. If the point has coordinates $(r,\theta)$ then it also has coordinates $(-r,\theta\pm\pi),$ $(r,\theta\pm2\pi),$ $(-r,\theta\pm3\pi),$ $(r,\theta\pm4\pi),$ and so forth. So if $r = f(\theta)$ is the formula of a curve then each of the following is also a formula of the same curve: \begin{align} -r &= f(\theta+\pi),\\ -r &= f(\theta-\pi),\\ r &= f(\theta+2\pi),\\ r &= f(\theta-2\pi),\\ \end{align} and so forth.