When computing the unit outward normal vector, for example on a surface of a lateral surface of a cylinder or sphere, I always go by the order the variables (right-hand rule in case of the cartesian coordinates) are used, i.e., $(\theta, z)$ if it's the lateral surface of a cylinder whose axis is the $z$-axis and $(\phi, \theta)$ if it's the surface of a sphere.
Is there an alternative explanation to this ? Thanks in advance.