If the range we desire is in centigrade +45 degrees (about scalding) to 0 (frozen) degrees it may be a possibility just like it was in the lethally hot Triassic period when we were 30,000 km nearer to the sun and had a globally arid and dry climate with a milder temperature gradient between poles. The creation of the new ocean that caused a circumpolar current eventually led to atmospheric currents that rotated from west to east. Atmospheric and oceanic currents stopped the transfer of warm, tropical air and water to the higher latitudes. As a result of the warm air and currents moving northward, Antarctica cooled down so much that it became frigid.
one pole needs to fluctuate say +20 + (winter) to +40 (summer)
whilst other fluctuates say 0 or less (winter) to +20 (a balmy summer)
North Pole Average summer 0° C Winter -40° C range = 40
South Pole Average Summer −28° C Winter −60° C range = 32
Today moving earth closer to the sun can conceivably move those values back by say +45
There are other factors needed to adjust some of the natural effects and in our solar system there are many still to be determined reasons that each planet has one pole significantly different to the other.
For one pole to be arid there would need to be almost all water concentrated near the opposite pole where it would needs be trapped as snow and ice much of the year with very little rain at the other arid pole. this is made more complex since heated air and water tend to cycle between hot areas and cold areas
creating prevailing winds.
The problem with this is that much of the planet would be inhospitable (especially at the equator) thus the central belt would also be arid and transportation (night) / living need a comparatively higher tech solution such as windmills and evaporators to produce cooling. Let me know where I can buy shares in the water company since they will be worth their weight in gold :-)
The only working alternative is to have a planet such as Proxima Centauri b with fixed "polar" rotation, see "tidal locking" just as our moon is locked towards us but in that case there would be NO night or day as 1/2 the planet would be hot in the sun and the opposing side cold so inhabitants would presumably be "poles apart" few would live there in perpetual darkness, but that could aid a story of conflict over energy sources.
An unusual reverse example, may be Tau Boötis, a star that is probably tidally locked to its planet Tau Boötis b However at an estimated surface temperature of 1600 degrees K is too hot to contemplate, unless we can find a "cooler" example.