Since one is to blow the Shofar when the wide opening is facing up, if in a particular case it faces sideways, does the Baal Tokea have to incline his head to comply with the above.

Indeed Poskim mention that the wide opening of the Shofar should be facing up, (Shulchan Aruch O.H, 585: 2, Mishna Berura 9). However, when it isn’t, Horav Dovid Pam Shlit’a and Horav Aharon Miller Shlit’a maintain that the Baal Tokea does not have to incline his head for it. Horav Shlomo Miller’s Shlit’a opinion is similar