  The two boundary Temperley-Lieb algebra $TL_k$ arises in the transfer matrix
formulation of lattice models in Statistical Mechanics, in particular in the
introduction of integrable boundary terms to the six-vertex model. In this
paper, we classify and study the calibrated representations---those for which
all the Murphy elements (integrals) are simultaneously diagonalizable---which,
in turn, corresponds to diagonalizing the transfer matrix in the associated
model. Our approach is founded upon the realization of $TL_k$ as a quotient of
the type $C_k$ affine Hecke algebra $H_k$. In previous work, we studied this
Hecke algebra via its presentation by braid diagrams, tensor space operators,
and related combinatorial constructions. That work is directly applied herein
to give a combinatorial classification and construction of all irreducible
calibrated $TL_k$-modules and explain how these modules also arise from a
Schur-Weyl duality with the quantum group $U_q\mathfrak{gl}_2$.
