  The LLT polynomials $\mathcal{L}_{\mathbf{\beta}/\mathbf{\gamma}} (X;t)$ are
a family of symmetric polynomials indexed by a tuple of (possibly
skew-)partitions $\mathbf{\beta}/\mathbf{\gamma}=
(\beta^{(1)}/\gamma^{(1)},\ldots,\beta^{(k)}/\gamma^{(k)})$. It has recently
been shown that these polynomials can be seen as the partition function of a
certain vertex model whose boundary conditions are determined by
$\mathbf{\beta}/\mathbf{\gamma}$. In this paper we describe an algorithm which
gives a bijection between the configurations of the vertex model with boundary
condition $\mathbf{\beta}/\mathbf{\gamma} =
(\beta^{(1)}/\gamma^{(1)},\beta^{(2)}/\gamma^{(2)})$ and those with boundary
condition $(\mathbf{\beta}/\mathbf{\gamma})_{swap} =
(\beta^{(2)}/\gamma^{(2)},\beta^{(1)}/\gamma^{(1)})$. We prove a sufficient
condition for when this bijection is weight-preserving up to an overall factor
of $t$, which in turn implies that the corresponding LLT polynomials are equal
up to the same overall factor. Using these techniques, we are also able to
systematically determine linear relations within families of LLT polynomials.
