In a word, no.
Technically, it inverts phase twice - each 12AX7 inverts, so final result is no inversion. The 12AU7 is a cathode follower and it maintains phase.
At the Hewlett building on Stanford campus, there's a reproduction of a letter from Bill Hewlett to Fred Terman on his new book. Bill was his student at the time. Bill talks about how good the book is, but notes that an amplification equation for a triode omitted a negative sign, as the signal is inverted. In a few years, he would figure out how to use a light bulb as a cathode resistor to apply compensating feedback in an audio oscillator and with that successful product, start HP with Dave Packard.