The current state-of-the-art in quadruped locomotion is able to produce robust motion for terrain traversal but requires the segmentation of a desired robot trajectory into a discrete set of locomotion skills such as trot and crawl. In contrast, in this work we demonstrate the feasibility of learning a single, unified representation for quadruped locomotion enabling continuous blending between gait types and characteristics. We present Gaitor, which learns a disentangled representation of locomotion skills, thereby sharing information common to all gait types seen during training. The structure emerging in the learnt representation is interpretable in that it is found to encode phase correlations between the different gait types. These can be leveraged to produce continuous gait transitions. In addition, foot swing characteristics are disentangled and directly addressable. Together with a rudimentary terrain encoding and a learned planner operating in this structured latent representation, Gaitor is able to take motion commands including desired gait type and characteristics from a user while reacting to uneven terrain. We evaluate Gaitor in both simulated and real-world settings on the ANYmal C platform. To the best of our knowledge, this is the first work learning such a unified and interpretable latent representation for multiple gaits, resulting in on-demand continuous blending between different locomotion modes on a real quadruped robot.