  When ${\bf t} := \langle t_1,t_2,\ldots,t_k\rangle$ is a sequence of
transpositions on the finite set $n:=\{0,1,\ldots,n-1\}$, then $\bigcirc{\bf
t}:= t_1\circ t_2\circ\cdots\circ t_k$ denotes the compositional product of the
sequence. Our paper treats the set ${\rm Prod}({\bf t})$ of all $\bigcirc{\bf
s}$, where ${\bf s}$ is a sequence obtained by rearranging the terms of ${\bf
t}$. The paper characterizes the set of all transpositional sequences ${\bf t}$
for which ${\rm Prod}({\bf t})$ is the subset of a single congugacy class in
the symmetric group ${\rm Sym}(n)$; we call such ${\bf t}$ {\it conjugacy
invariant}. At the opposite extreme, the paper studies conditions under which
${\bf t}$ is {\it permutationally complete}, which is to say, those ${\bf t}$
for which either ${\rm Prod}({\bf t}) = {\rm Alt}(n)$ or ${\rm Prod}({\bf t}) =
{\rm Sym}(n)\setminus{\rm Alt}(n)$.
