Can we eliminate adjacent opposing pairs while classifying the surface in other than the second step?

Question

When we apply the method of classification of surfaces to a given surface, in the second step we can eliminate adjacent opposing pair, but can we do this when, for example, in the sixth step if we get any adjacent opposing pair like $$.. d d^{-1}... ?$$

Note that, by saying that ...th step, I'm referring to the constructive proof of the classification of closed & compact surfaces theorem in Kinsey's Topology of Surfaces book.

Answer 1

Yes, you are taking a connect sum with a sphere, which topologically does not change your surface.

Answer 2

Although this question is answered already, I'm gonna add this answer just for some future people.

Please have a look at Massey's Algebraic Topology: An Introduction particularly, Theorem 5.1 (the classification theorem for compact surfaces). Just see the four pictures on page 22. The pair of edges mentioned in this question is a pair of "First Kind" (Massey's terminology). In step 2 of Massey's proof, pictorially you'll see how we can eliminate such a pair. I wish I could draw those four pictures.