Metrologist, that is a QFH. The reason why they can be oriented in any direction is because the direct signal from the sky is RHCP and a QFH antenna naturally selects for RHCP signals and has a substantial rejection of LHCP nomatter what direction it is coming from. (The pattern of QFH antennas can be tweaked quite a bit, a tall narrow one will have more gain towards the horizon, good for amateur and especially APT satellite reception, a squatter QFH will have a rounder pattern and more gain vertically).
Reflections because they are a mirror image reverse polarization. This is a very useful property and can be used for interesting things like determining the water content of the ground by ratio of reflected vs direct GPS signals over it.
Patch antennas ability to reject LHCP signals is dependent on the angle of incidence to the patch, at low angles (off to the side) the rejection is very poor. They basically become useless as far as rejection at low angles. This is why its smart to set your GPS software to reject low angle sats for highest accuracy in urban canyons (where the reflections are severe) , but the need to do that varies depending on what kind of antenna you use.
Odd reflections are canceled while even reflections add.
I have a PCTel (originally made by Lucent) timing antenna that looks like a soft serve ice cream cone or missile nose cone in shape and it has a QFH inside it that looks much like yours. (Have not opened it, just seen pictures)