What would happen if the North and South Poles were arid deserts and the Equator was frigid zone?

First, that is impossible.
Second, Antarctica is is already an arid desert. It has less precipitation than any other continent.

The Antartic continent is arid because the surrounding oceans and atmosphere are too cold to send water vapour across the land. The blizzards it has are of ancient snow being blown about, rather like dust storms in tropical deserts.

(A "desert" is a region of extremely low mean annual precipitation,  not of high temperature, but I can't remember the limit used for the definition. So the Antarctic is a desert even though very cold.)

The situation you ask would be impossible because the general temperature-ranges are set by the angle at which they are irradiated by the Sun; and the angle of the Earth's axis to the plane of its orbit round the Sun. So even if the Earth was at 90º to its axis, putting what we know as the Poles at latitude 0º, it would still have a hot Equator and cold Poles.