It turns out there's something very much like this, right next door to Earth. Observe the topography of Mars:
(Image courtesy Wikipedia; larger version here.)
The large blue areas in the northern hemisphere, making up about a third of the surface, are about 4 to 6 km lower in altitude than the yellow/orange highlands to the south. This dramatic discrepancy is known as the Martian dichotomy and there are two major schools of thought for how it came about.
The first is that the dichotomy represents one or more colossal impact craters. If it was from a single impactor, it would be the largest impact crater in the solar system. Opinions are divided as to whether a single impact can explain the resulting geography, or whether it's better modeled as multiple (still massive) impacts. Of course you could do either.
The other theory is related to long-ago Martian tectonics. The theory goes that for reasons unknown (and this is still an active subject of research on Earth, let alone Mars), one hemisphere featured one or more huge upwellings of material from the mantle into the crust, and the other featured downwellings. Over an extremely long period of time, the result is that crustal material migrates from one hemisphere to the other. (The average thickness of the southern crust is twice that of the north.)
In the case of Mars, the dichotomous hemispheres are north and south, but this may just be coincidence. Neither process has any particular reason to favor one orientation over another as far as I know, so having a dichotomy between east and west hemispheres should be possible.
Note that there are still local variations: there are craters and mountains in the north part of Mars, valleys in the south part. However, the difference should be enough to establish the broad climate dichotomy that you're looking for.