The online math textbook accompanying the Precalculus course I'm taking provides methods in the form of basic substitution tests to guarantee whether each symmetry necessarily holds and posits that if it fails the test(s) for a particular symmetry then it is inconclusive (the symmetry being tested for may still exist) but that if it passes at least one of the respective tests then that symmetry is guaranteed to apply. They give a graphic counter-example of an equation that fails tests for a symmetry but still is, but with no explanation as to why (It's $r=θ+2π$ restricted to interval $-4π≤θ≤0$, under which constraint it appears to be maybe half a circuit plus an endpoint instead of a full loop, but I don't know how or if that pertains generally.)
I am wondering A] if there is a more formal way to describe the method conveyed by these tests as a relation (beyond how they explained it) and B] what other analytical methods are there definitively to determine whether or not a polar equation has symmetry (or perhaps how to determine presences and by extension absences of any/all symmetries generally, other than the polar equivalents of x-axis and y-axis and origin in cartesian grid oriented standard), and C]for any deeper understanding.
Below is their text description of testing for symmetry with respect to polar axis. (Their other two descriptions follow the same form.)
"If replacing (r, θ) by (r, -θ) or (-r, π-θ) results in an equivalent equation, the polar graph is symmetric with respect to the polar axis."
They proceed to give two example equations applying all the six tests for determining any conclusivity of symmetry based upon that, but don't add any rigor or fill a non-graphic intuition gap.
I have no clue what other analytical methods exist to determine conclusively that a symmetry does not hold, but I have an idea how to rephrase the conditional assertions more logic-ly (though it is probably improper or messy or longer than necessary). Concerning symmetry w.r.t polar axis: If it holds, then
$∀\{(r, θ)\}∈f(\{r, θ\}): \{(r, - θ)∨(r, -θ+π)\}∈f(\{r, θ\})$
where f({r, θ}) is the function relating r to theta and (r, θ) is a polar coordinate r comma theta.