If you say it a bit so there isn't a lot of water in it, it isn't necessary to hang the brush with the bristles down. Any remaining moisture in a nearly dry brush evaporates via capillary action.Rinse under warm running water, gently squeeze, repeat until water runs clear. I move the brush in all directions under the water. After the final rinse/squeeze, I flick it until no water comes out, then strop on my bath towel in various directions. If I look at the knot and the fibers are bunching up, I'll comb it with my fingers. The whole process takes less than two minutes.
Here is where opinions differ.... hang in a stand knot down or set it on the handle knot up?
I split the difference. I put it knot down in a stand for a day, then on the handle for three days (4 brush rotation).
If you use synthetic brushes, dryng time is quite short and it becomes even less important how you leave the brushes between shaves.