  Connections between set-theoretic Yang-Baxter and reflection equations and
quantum integrable systems are investigated. We show that set-theoretic
$R$-matrices are expressed as twists of known solutions. We then focus on
reflection and twisted algebras and we derive the associated defining algebra
relations for $R$-matrices being Baxterized solutions of the $A$-type Hecke
algebra ${\cal H}_N(q=1)$. We show in the case of the reflection algebra that
there exists a ``boundary'' finite sub-algebra for some special choice of
``boundary'' elements of the $B$-type Hecke algebra ${\cal B}_N(q=1, Q)$. We
also show the key proposition that the associated double row transfer matrix is
essentially expressed in terms of the elements of the $B$-type Hecke algebra.
This is one of the fundamental results of this investigation together with the
proof of the duality between the boundary finite subalgebra and the $B$-type
Hecke algebra. These are universal statements that largely generalize previous
relevant findings, and also allow the investigation of the symmetries of the
double row transfer matrix.
