Does this algebra whose signature is (1,1) have a name?

Question

Let there be an algebra $(S,f,t)$ with the laws:

$$ f(t(x)) = t(x) \\ t(f(x)) = t(x) $$

or, put another way,

$$ f \circ t = t \\ t \circ f = t. $$

1. Does that particular algebra have a name?
2. Does a set with two unary operations (algebras whose signature is (1,1) in general) have a name?

The closest thing I found (by replacing $f$ with $x$ and $\circ$ with $\vee$) was null semigroup such as $(S,\vee,t)$ where

$$ x \vee t = t \\ t \vee x = t $$

(but without the idempotency law $x \vee x = x$, in case $\vee$ reminds you of lattices). But the signature of such null semigroup is (2,0), not (1,1).