Just do it without. Symmetry for Calc III problems are harder to detect than regular Calc I or II integrals. It makes things so much easier.
To prove symmetry you need to make sure that the domain and the integrand is symmetric. If it is not, then you are screwed and just have to do it the long ways anyways.
So to save you time, especially on timed tests and quizzes don't worry about it and just do the problem.