  Many discrete mathematics problems in phylogenetics are defined in terms of
the relative labeling of pairs of leaf-labeled trees. These relative labelings
are naturally formalized as tanglegrams, which have previously been an object
of study in coevolutionary analysis. Although there has been considerable work
on planar drawings of tanglegrams, they have not been fully explored as
combinatorial objects until recently. In this paper, we describe how many
discrete mathematical questions on trees "factor" through a problem on
tanglegrams, and how understanding that factoring can simplify analysis.
Depending on the problem, it may be useful to consider a unordered version of
tanglegrams, and/or their unrooted counterparts. For all of these definitions,
we show how the isomorphism types of tanglegrams can be understood in terms of
double cosets of the symmetric group, and we investigate their automorphisms.
Understanding tanglegrams better will isolate the distinct problems on
leaf-labeled pairs of trees and reveal natural symmetries of spaces associated
with such problems.
