  We study the decomposability of a finite Blaschke product $B$ of degree $2^n$
into $n$ degree-$2$ Blaschke products, examining the connections between
Blaschke products, the elliptical range theorem, Poncelet theorem, and the
monodromy group. We show that if the numerical range of the compression of the
shift operator, $W(S_B)$, with $B$ a Blaschke product of degree $n$, is an
ellipse then $B$ can be written as a composition of lower-degree Blaschke
products that correspond to a factorization of the integer $n$. We also show
that a Blaschke product of degree $2^n$ with an elliptical Blaschke curve has
at most $n$ distinct critical values, and we use this to examine the monodromy
group associated with a regularized Blaschke product $B$. We prove that if $B$
can be decomposed into $n$ degree-$2$ Blaschke products, then the monodromy
group associated with $B$ is the wreath product of $n$ cyclic groups of order
$2$. Lastly, we study the group of invariants of a Blaschke product $B$ of
order $2^n$ when $B$ is a composition of $n$ Blaschke products of order $2$.
