Etymology - אף על פי

Consider the phrase אף על פי meaning literally "nose on my mouth" and being used ubiquitously as "even though" as in the sense even though you would have thought X to be the decisive factor, in fact it is Y.  How does even though derive from nose on my mouth?

Consider the use of the possesive form -- "my mouth"  (פי) -- as opposed to the plain word of  "a nose" אף. Consider also the defining of the nose in relation to the mouth and not the other way around -- פי מתחת אף.

Those points mentioned I think the answer is thus: the phrase of אף על פי having the meaning of even though built into the very words themselves:   the nose, naturally being the greater of the two organs by virtue of it's being elevated above the mouth, is nonetheless subordinate/subservient to פי. Even though you would have thought the nose should be the primary -- אף על פי כן -- primary focus is in fact given to the פה.

והבן היטב.