Dry-ness won't stay the same due to differing wind cycles
The Earth, due to its rotation, generally has three 'cells' of circulating air. The Hadley cell, Mid-latitude cell, and Polar cell. Due to the patterns of rising and falling air, these cells cause the same general climate patterns to emerge around the Earth (assisted in large part by oceanic currents). I'll describe some of the general effects briefly.
The Hadley cell generally causes wet conditions at the equator and dry conditions at 30 degrees. You can see this on each continent with the tropical rainforests near the equator and the worldwide bands of desert around 30 N and S.
The Mid-latitude cell generally features Mediterranean climates (wet winters, dry summers) on West coasts of continents at lower latitudes, but cool wet climates on West coasts at higher latitudes. Examples are Spain vs. England; Los Angeles vs. Seattle; Santiago vs. Valdivia in Chile. The East coasts of continents in the Mid-latitude cell are generally much hotter and wetter in the summer. Compare the summer conditions of San Francisco and Washington DC, or Lisbon and Seoul.
The Polar cell is generally devoid of moisture. Rainfall north of 60 degrees is generally very low; for example look at Yakutsk, Russia or Barrow, Alaska.
The polar ice caps are dry! In fact, Barrow is significantly drier than Tibet. Yet it still snows there every month of the year, and sees almost a meter of snow a year. Antarctica is drier still, and has miles of ice in places.
All in all, your poles are going to be dry, but as long as they are cold enough, they are going to develop ice caps over time.