I do for the first couple days after cleaning so that any oil in the barrel drips out the muzzle. After a couple days I flip it around,  mainly because they fit better in the safe

Good idea with muzzle loaders. Oil will get in to the nipple and sometimes its hard to get cleaned out after storage.

I store most rifles muzzle down.  If not, they are horizontal with a slight muzzle down pitch.  I’m generous with bore lube and don’t want it creeping into the action or wood.  Most of my cases are hard plastic with foam interiors so I put a small absorbent cloth near the muzzle to capture any excess.  Been doing this for years and no bad effects. I try to get them all out a couple times a year for an inspection if they have not been used.