How to create temperature gradient E-W rather than N-S?

[Triplicate]

Do you care if the world as mapped is all there is? You could get the straight lat/Lon on the inside of a ring world, only a reasonable e-w would be manymany times the n-s dimension.

I recall trying hard to determine whether this map was the entire world. At first, I couldn't tell the scale, because many things about terrain are fractal. But the climate variation suggested a large-scale map. And, in retrospect, the size of mountain ranges. It's entirely possible the polar regions were truncated, as on many Mercator maps.
An artificial world doesn't fit the setting/style I had in mind.

A world with a 90 degree inclination would have some interesting day/night/season patterns.

I've never spent a lot of time thinking about this, but...
As I see it, a world with a high inclination would indeed have extreme seasons. However, it would still have one pole toward the sun at one time and the other at another time. I don't see why it would have a large difference from one hemisphere to another.

Elevation and latitude would be some key points to consider. For instance, the many countries of the western hemisphere are mainly near the equator and nowhere on any of them are there mountains higher than 5,000 feet. .... There's little land north or south of the tropics

Problem with those ideas: This world was something like 90% land. The specific feature I'm trying to replicate is the east being much colder than the west at the same latitude.
I hadn't really thought about using altitude to change climate.

[atpollard]

The specific feature I'm trying to replicate is the east being much colder than the west at the same latitude.

I don’t think that this is possible without approaching tidal lock. If the world rotates on a North-South axis in anything approaching 24 hours, the East and West will alternate day/night too rapidly to have a significant difference in temperature … the atmosphere will conduct heat and moderate the temperatures in horizontal bands.

A very slow rotation – like a 700 hour day – will allow the east or west hemisphere to experience a one week, all daylight summer, a one week slow sunset fall, a one week, all night winter and a one week slow sunrise spring.

With respect to the 90 degree inclination, the world rotates on a polar east-west axis (with north-south being defined by the celestial plane of the sun and planets). Assuming 24 hour rotation day and 365 day revolution year … The East polar hemisphere will experience 3 months of Tropical/Desert no sunset Summer, followed by 3 months of temperate sunrise-sunset fall, followed by 3 months of Arctic darkness Winter, followed by 3 months of temperate sunrise-sunset spring. During the Spring and Fall, the East and West Hemispheres will diverge towards opposite seasons as your map indicates (without the eternal day/night of tidal lock).

To heat only one side of a rotating sphere involves a minor heat source orbiting in a nearly geo-stationary orbit … a micro-sun orbiting over the east hemisphere. The micro-sun would need to advance slowly around the globe to move the ‘seasons’ east-west, otherwise one hemisphere gets perpetual summer and the other perpetual winter. I think that this starts to move from sci-fi into fantasy (not a criticism, just an observation) ... but Ptolemy would approve.

[Triplicate]

To heat only one side of a rotating sphere involves a minor heat source orbiting in a nearly geo-stationary orbit … a micro-sun orbiting over the east hemisphere. The micro-sun would need to advance slowly around the globe to move the ‘seasons’ east-west, otherwise one hemisphere gets perpetual summer and the other perpetual winter.

Which is basically what I had in mind with the brown dwarf/star combination. The whole point is to have one hemisphere in perpetual "summer" and the other in "winter".

At first, I was thinking this might be a fairly common type of planet in the galaxy. Hoping there might even be a known star system with suitable parameters, I found this instead:
https://en.wikipedia.org/wiki/Brown-dwarf_desert
Brown dwarfs are rare in close orbits around stars. The exceptions are where the star is a red dwarf.
That is, systems where the same planet could receive significant heating from both are unlikely.