  Building on a model introduced by Mandal and Jarzynski [Proc. Natl. Acad.
Sci. U. S. A., {\bf 109}, (2012) 11641], we present a simple version of an
autonomous reversible Maxwell's demon. By changing the entropy of a tape
consisting of a sequence of bits passing through the demon, the demon can lift
a mass using the coupling to a heat bath. Our model becomes reversible by
allowing the tape to move in both directions. In this thermodynamically
consistent model, total entropy production consists of three terms one of which
recovers the irreversible limit studied by MJ. Our demon allows an
interpretation in terms of an enzyme transporting and transforming molecules
between compartments. Moreover, both genuine equilibrium and a linear response
regime with corresponding Onsager coefficients are well defined. Efficiency and
efficiency at maximum power are calculated. In linear response, the latter is
shown to be bounded by 1/2 if the demon operates as a machine and by 1/3 if it
is operated as an eraser.
