  For any polyhedral norm, the bisector of two points is a polyhedral complex.
We study combinatorial aspects of this complex. We investigate the sensitivity
of the presence of labelled maximal cells in the bisector relative to the
position of the two points. We thereby extend work of Criado, Joswig and Santos
(2022) who showed that for the tropical distance function the presence of
maximal cells is encoded by a polyhedral fan, the bisection fan. We initiate
the study of bisection cones and bisection fans with respect to arbitrary
polyhedral norms. In particular, we show that the bisection fan always exists
for polyhedral norms in two dimensions. Furthermore, we determine the bisection
fan of the $\ell_{1}$-norm and the $\ell_{\infty}$-norm as well as the discrete
Wasserstein distance in arbitrary dimensions. Intricate combinatorial
structures, such as the resonance arrangement, make their appearance. We apply
our results to obtain bounds on the combinatorial complexity of the bisectors.
