Do we need to check for all kinds of symmetry in a compound to declare it achiral?
We need to check all 4 symmetries to determine whether a compound is optically active or not.
Consider the following compound. It has only Alternating Axis of Symmetry

Note that the rings are in perpendicular planes. The following diagrams illustrate the top view of the molecule.The stereochemistry of the methyl groups can be:

$S_4$ axis of symmetry is seen only in the following diastereomer where the methyl groups are anti to each other in each ring and the view looks like a Nazi cross from the top.