Test for symmetry for polar graphs From a calculus book I'm reading: "Unlike the graphs of an equation in $x$ and $y$, the graph of an equation $r=f(\theta)$ can be symmetric with respect to the polar axis, the line $\theta = \pi/2$, or the pole without satisfying one of the tests for symmetry. This is true because of the many different ways of specifying a point in polar coordinates."
Why is the previous paragraph true? Why are there many ways to specify a point in polar coordinates and how does that affect the symmetry test?


The origin has infinitely many "names" $(0,\theta)$ where $\theta$ can be anything. And every point $(r,\theta)$ could also be called $(r,\theta+2n\pi)$, $n$ any integer. How that affects symmetry tests is unclear: it should not affect a properly constructed symmetry test. –  André Nicolas on July 27, 2013, at 7:06 

André has posted what I think is a satisfactory answer in the comments, hence this wiki answer.
