Handles and cross-handles are equivalent in the presence of a cross-cap.
Dyck's Theorem
See also
Cross-Cap, Cross-Handle, Handle, von Dyck's TheoremExplore with Wolfram|Alpha
References
Dyck, W. "Beiträge zur Analysis situs I." Math. Ann. 32, 459-512, 1888.Francis, G. K. and Weeks, J. R. "Conway's ZIP Proof." Amer. Math. Monthly 106, 393-399, 1999.Referenced on Wolfram|Alpha
Dyck's TheoremCite this as:
Weisstein, Eric W. "Dyck's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DycksTheorem.html