We present the results of millimetre (33 and 35 GHz) and centimetre (2.1, 5.5 and 9.0 GHz) wavelength observations of the neutron star X-ray binary Circinus X-1, using the Australia Telescope Compact Array. We have used advanced calibration and deconvolution algorithms to overcome multiple issues due to intrinsic variability of the source and direction dependent effects. The resulting centimetre and millimetre radio maps show spatially resolved jet structures from sub-arcsecond to arcminute angular scales. They represent the most detailed investigation to date of the interaction of the relativistic jet from the X-ray binary with the young supernova remnant in which it is embedded. Comparison of projected jet axes at different wavelengths indicate significant rotation of the jet axis with increasing angular scale. This either suggests interactions of the jet material with surrounding media, creating bends in the jet flow path, or jet precession. We explore the latter hypothesis by successfully modelling the observed jet path using a kinematic jet model. If precession is the right interpretation and our modelling correct, the best fit parameters describe an accreting source with mildly relativistic ejecta ($v = 0.5 c$), inclined close to the plane of the sky ($i = 86^{circ}$) and precessing over a 5-year period.