Actually, in a world where one side was very warm (burning) and the other very cold (completely frozen over), there would be a huge heat dissipation in the form of stormwinds between the two. The whole region where you place a civilization would constantly be blazed by storms the size of continents - not a very nice place to live indeed!
Apart from that, if I understand your compass rose correctly, the part of the world you're centering on is the north pole. If that's the case, all directions outward would be south, not just two of the four. In fact, going west would mean you'd be going in a clockwise circle around the north pole, going east would be going around in a counterclockwise fashion. Since there really isn't a north pole on a tidally locked world (there being no axis around which the planet revolves), I think a more logical separation would be something like Night (the freezing side of the planet), Day (the burning side), Dusk and Dawn (both directions perpendicular to the Night-Day line). That would lead to awesome expressions like "Ranunculus was a two-day trip Dawnward from Lotos, with amazing views Nightward on the Sea of ... (etc)", or "the Duskbound ship leaves port in three hours!" *
All in all, I really like your map, and the concept of eternal night, eternal day better than my remarks above might seem to say, and I hope some of those remarks are helpful!
(* what I come to think of now is: how do you measure time on a world that doesn't revolve? It'd only make one rotation around its axis in a year's time, so even the stars would be useless! Interesting to look into, right?)