Sundogs flank the sun.  Beneath them and the horizon, not quite mirror images, subtly different – subparhelia – flank a subsun.

All these halos involve reflections.  

Sundogs and subparhelia are very similar. They form when sun rays pass between vertical side faces of horizontal plate crystals inclined 60° to each other.  But their paths through the ice are not smooth.  The light reflects up and down between the large hexagonal top and bottom faces.    

An even number of reflections and the sun ray exits downwards to contribute to a sundog.  

An odd number and the ray emerges upwards to count towards a subparhelion seen only from an aircraft or mountainside.