The gravitomagnetic clock effect and the Sagnac effect for circularly rotat
ing orbits in stationary axisymmetric spacetimes are studied from a relativ
e observer point of view, clarifying their relationships and the roles play
ed by special observer families. In particular, Semerak's recent characteri
zation of extremely accelerated observers in terms of the two-clock clock e
ffect is shown to be complemented by a similarly special property of the si
ngle-clock clock effect.