In Shorinji Kempo we make a point of learning to use different hip positions. I forget the exact Japanese names but basically we have hip forwards and hip backwards stances that we use in different situations.
Shorinji Kempo has an unusual herritage that is a mixture of Japanese Jujitsu and Chinese Qi Na (from Northern Shaolin style). For this reason I don't know what the origin of this emphasis on hip position actually is. It might even be a kenjutsu type skill.
One principle applied in bagua striking is to use the spine as a whip to transmit power. The whipping motion starts as a wave in the hips; any strike using this principle works through a range of hip pitch angles. This video is a sufficient example.
This is a slightly iffy answer in that it's basically drawing in individual techniques from two different styles (and maybe a non-MMA movement), but boxing has the bolo punch, which was introduced by Filipino workers who sourced the movement from a combination of their native styles and the machete movement used to cut sugar cane. While most people think of the circular movement of the arm as the fundamental part of it, the starting location by the hips is actually to use the hip movement to drive the cut without having to rely on arm strength.
