Although I'm a quite avid atmospheric optics observer, I rarely have to look down to see such phenomena. However, on a flight I took not too long ago, I was fortunate enough to see a few rare halos.
The interesting thing is that the subhorizon halos are like a reflection of the those above the horizon, and the very cool thing about it is that this reflection actually takes place inside the ice crystals that create these optical phenomena in the first place (on a lower horizontal plate). However, the sub- and over-horizon halos are not perfect reflections of each other.
The image below shows the antisolar point (bright spot in the lower right of the image), a portion of the subparhelic circle going horizontally through the antisolar point and the parry antisolar arcs, seen as two lines going through the antisolar point at an angle of about 60 degrees from the horizontal.
Les Cowley's excellent website does a much better job at explaining this stuff.
I'm just very happy that I saw them: