Yes, it is possible, although the flavor would probably be wrong for what you have in mind.
Climate varies with latitude and altitude. Mountains on the equator have permanent glaciers.
So to have the equator closed by ice you just have to make it high enough.The easiest way would probably be to make the planet less spherical. This has the benefit of not needing mountains, although you'd need to keep the equator free of of oceans, ie. have a supercontinent that encircles the planet on the equator.
This state is not stable as planets are by definition in hydrostatic equilibrium. So you'd need some reason, such as divine or alien interference that explains why the planet is less spherical than it should. Planet had too fast rotation and the gods fixed it, maybe?
Alternately, since you need the supercontinent anyway, you can just fiat that every point on the equator just happens to be high enough thanks to various highlands and mountain ranges. There is no real reason why not. Note that since the equator would have much less heat absorbing ocean and much more heat reflecting ice and snow, it would not have to be as high as is in our world for permanent glaciers. There would be permanent winds away from the glaciers and without ocean currents, there would really be no way for heat from surrounding warmer lands to melt the equator.
Making the poles hot is more problematic. With huge glaciers and no seas on the equator it is perfectly reasonable to assume mostly dried out ocean basins surrounding the poles. If you configured what remains of the seas for efficient heat transport, I'd guess there is nothing stopping the poles from being hot.
So you'd have a polar ocean with water heated closer to the equator. There would be islands and coastal lands with geology largely formed of marine evaporites.
The problem with this solution is that it would do nothing about the poles being short on sunlight at least half the year. So the agricultural productivity and supported population levels would not be that good.
Then again, I doubt it is that crucial to you to have huge farmlands on the poles? Since you need polar oceans anyway for heat transport, you can place only islands with hardy fishermen on the poles and have the agricultural lands on the coasts closer to the equator with better sunlight.