Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE PLANE 469
(a) If a surface is symmetric in each of two coordinate
planes, it is symmetric in the coordinate axis in which the two
planes meet.
(b) If a surface is symmetric in each coordinate plane, it is
symmetric in the origin.
(c) If a surface is symmetric in a coordinate plane and in
the coordinate axis perpendicular to this plane, it is symmetric
in the origin.