  In this paper we develop a complete theory of factorization for isometries of
hyperbolic 4-space. Of special interest is the case where a pair of isometries
is linked, that is, when a pair of isometries can be expressed each as
compositions of two involutions, one of which is common to both isometries.
  Here we develop a new theory of hyperbolic pencils and twisting planes
involving a new geometric construction, their half-turn banks. This enables us
to give complete results about each of the pair-types of isometries and their
simultaneous factorization by half-turns. That is, we provide geometric
conditions for each such pair to be linked by half-turns. The main result gives
a necessary and sufficient condition for any given pair of isometries to be
linked. We also provide a procedure for constructing a half-turn linked pair of
isometries of $\mathbb{H}^4$ that do not restrict to lower dimensions, yielding
an example that gives a negative answer to a question raised by Ara Basmajian
and Karan Puri.
