The typical kaleidoscope has three mirrors in the shape of a triangular prism. One angle between the mirrors is 90 degrees. If the smallest angle between the mirrors is equal to 360 degrees divided by an even number, then we get an image with rotational symmetry. Usually, this symmetry is not compatible with translational symmetry, resulting in images which are not periodic. Also, the rotational symmetry is destroyed far away from the mirrors.
We can see this nicely for an angle of 22.5 degrees between mirrors. This gives us a local eight-fold rotational symmetry. But after multiple reflections the images do not fit together anymore:
Similarly an angle of 36 degrees gives local 5-fold rotational symmetry: