Key results of the theory of Hamiltonian systems rely on the twist condition, i.e., the assumption of a monotonic profile for the frequency of orbits. Many physical systems of great importance have a nonmonotonic profile, so their Hamiltonian models violate the twist condition, leading to characteristic phenomena such as reconnection-collision sequences of periodic orbits and shearless curves, which represent robust transport barriers in phase space. Recently, the so-called Biquadratic Nontwist Map was used to study nontwist systems with multiple shearless curves and the transition to chaos. However, some local processes, involving periodic orbits, that occur in this new map have not been studied yet. In this work, we use the bifurcation theory to characterize local and global processes, like periodic orbit collision and the emergence of multiple isochronous island chains found in the Biquadratic Nontwist Map. Finally, due to its symmetry properties and the large number of bifurcations found, the results suggest that the Biquadratic Nontwist Map may be used to study bifurcations in area-preserving maps.