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