  In this paper we present some consequences of the description of matrix
representations of asymmetric truncated Toeplitz operators acting between
finite-dimensional model spaces. In particular, we prove that these operators
can be characterized using modified compressed shifts or the notion of shift
invariance. We also describe rank-one asymmetric truncated Toeplitz operators
acting between finite-dimensional model spaces. This paper is a sequel to the
paper of J. Jurasik and B. {\L}anucha "Asymmetric truncated Toeplitz operators
on finite-dimensional spaces".
