Yes, lateral inversion takes place in both plane and spherical mirrors.
In the case of a plane mirror, the size of the image is the same as the size of the object, and the image is laterally inverted. But in the case of a spherical mirror, the size of the image is not the same as the size of the object.
For example, when we raise our right hand in front of a plane mirror, our image seems to raise the left hand. The size of the image is the same as the size of the object. The same happens in the case of a spherical mirror, but the size of the image is not the same as the size of the object.