  For any positive integers $m$ and $k$, existing literature only determines
the number of all Euclidean self-dual cyclic codes of length $2^k$ over the
Galois ring ${\rm GR}(4,m)$, such as in [Des. Codes Cryptogr. (2012)
63:105--112]. Using properties for Kronecker products of matrices of a specific
type and column vectors of these matrices, we give a simple and efficient
method to construct all these self-dual cyclic codes precisely. On this basis,
we provide an explicit expression to accurately represent all distinct
Euclidean self-dual cyclic codes of length $2^k$ over ${\rm GR}(4,m)$, using
combination numbers. As an application, we list all distinct Euclidean
self-dual cyclic codes over ${\rm GR}(4,m)$ of length $2^k$ explicitly, for
$k=4,5,6$.
