A single-satellite dish (Dish 300, or original DTV style) will only
receive one satellite at a time (without having to re-aim). It works
fine with DTV or Dish.

A dual-satellite dish (like Dish 500) will be able to receive from two
different satellites without re-aiming.

A triple-satellite dish (new DTV style) only works for DTV as a triple
satellite dish. It works for Dish as only a single-satellite dish.
(Actually, DTV's original 3-satellite dish can work as a 2-satellite
dish for Dish if you move the LNBs around and change out the switch.)

With regard to multi-satellite dishes, the dishes work fine with either
service, but the 110-DTV LNB is weird, and the switches are not compatible
across the two services.

This may all change in the future.