  We investigate an operator on classes of languages. For each class $C$, it
outputs a new class $FO^2(I_C)$ associated with a variant of two-variable
first-order logic equipped with a signature$I_C$ built from $C$. For $C =
\{\emptyset, A^*\}$, we get the variant $FO^2(<)$ equipped with the linear
order. For $C = \{\emptyset, \{\varepsilon\},A^+, A^*\}$, we get the variant
$FO^2(<,+1)$, which also includes the successor. If $C$ consists of all Boolean
combinations of languages $A^*aA^*$ where $a$ is a letter, we get the variant
$FO^2(<,Bet)$, which also includes "between relations". We prove a generic
algebraic characterization of the classes $FO^2(I_C)$. It smoothly and
elegantly generalizes the known ones for all aforementioned cases. Moreover, it
implies that if $C$ has decidable separation (plus mild properties), then
$FO^2(I_C)$ has a decidable membership problem.
  We actually work with an equivalent definition of \fodc in terms of unary
temporal logic. For each class $C$, we consider a variant $TL(C)$ of unary
temporal logic whose future/past modalities depend on $C$ and such that $TL(C)
= FO^2(I_C)$. Finally, we also characterize $FL(C)$ and $PL(C)$, the
pure-future and pure-past restrictions of $TL(C)$. These characterizations as
well imply that if \Cs is a class with decidable separation, then $FL(C)$ and
$PL(C)$ have decidable membership.
