The examination of two 2010 Chandra ACIS exposures of the Circinus galaxy resulted in the discovery of two pulsators: CXO J141430.1-651621 and CXOU J141332.9-651756. We also detected 26-ks pulsations in CG X-1, consistently with previous measures. For ~40 other sources, we obtained limits on periodic modulations. In CXO J141430.1-651621, which is ~2 arcmin outside the Circinus galaxy, we detected signals at 6120(1) s and 64.2(5) ks. In the longest observation, the source showed a flux of ~1.1e-13 erg/cm^2/s (absorbed, 0.5-10 keV) and the spectrum could be described by a power-law with photon index ~1.4. From archival observations, we found that the luminosity is variable by ~50 per cent on time-scales of weeks-years. The two periodicities pin down CXO J141430.1-651621 as a cataclysmic variable of the intermediate polar subtype. The period of CXOU J141332.9-651756 is 6378(3) s. It is located inside the Circinus galaxy, but the low absorption indicates a Galactic foreground object. The flux was ~5e-14 erg/cm^2/s in the Chandra observations and showed ~50 per cent variations on weekly/yearly scales; the spectrum is well fit by a power law ~0.9. These characteristics and the large modulation suggest that CXOU J141332.9-651756 is a magnetic cataclysmic variable, probably a polar. For CG X-1, we show that if the source is in the Circinus galaxy, its properties are consistent with a Wolf-Rayet plus black hole binary. We consider the implications of this for ultraluminous X-ray sources and the prospects of Advanced LIGO and Virgo. In particular, from the current sample of WR-BH systems we estimate an upper limit to the detection rate of stellar BH-BH mergers of ~16 events per yr.
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