shaved his upper lip every morning three times with, three times against the grain

[Michael Bailey]

elided (understood) "when he was breaking in"

when he was using new blades for the first few times, he would
inevitably draw blood

(this is from my experience, anyway, a new Bic or whatever, or in the
old days a new razor blade would always draw blood the first few
shaves - not that I shave 3 times each way or ever have)

thus ---
Mucho shaved his upper lip every morning three times with, (and) three times
against the grain (in order) to remove any remotest breath of a
moustache, (and, hard on his skin as this routine was everyday, he
found that when using) new blades he drew blood invariably but kept at
it.