  We present MSO and FO logics with predicates `between' and `neighbour' that
characterise various fragments of the class of regular languages that are
closed under the reverse operation. The standard connections that exist between
MSO and FO logics and varieties of finite semigroups extend to this setting
with semigroups extended with an involution. The case is different for FO with
neighbour relation where we show that one needs additional equations to
characterise the class.
