A visitor asked me why are the mouths of the lion statues, the ones we see in front of numerous bank offices, have one mouth open and one closed.

One is male, and one is female, the different poses will in some way distinguish between the two. The one standing on the ball is the male one.

Answers to mouth question here en.wikipedia.org/wiki/Chinese_guardian_lions scroll down to 'appearance'

Sometimes the female has her mouth closed, and the male open. This symbolizes the enunciation of the sacred word "om". However, Japanese adaptions state that the male is inhaling, representing life, while the female exhales, representing death. Other styles have both lions with a single large pearl in each of their partially opened mouths. The pearl is carved so that it can roll about in the lion's mouth but sized just large enough so that it can never be removed.

or, times ago
female have to keep mouth close and care of the baby, which she pats with her left paw
male have to roar,and rul the wold, on which he pose his right pow. /-:

that means "we eat, and never spit out"

we chew you up and spit you out?

I always thought one was meant to be fierce (猛) while the other was benevolent (仁)