Two measurable bijections of a standard probability space are orbit equivalent if they have the same orbits up to Belinskaya » target= »_blank » title= »theorem. 
Belinskaya »>theorem. 
Belinskaya proved that orbit equivalence with integrable cocycles boils down to flip-conjugacy. We prove that it is optimal for all the odometers, namely for every odometer, we find a odomutant which is almost-integrably orbit equivalent to it but not flip-conjugate. This yields an extension of a theorem by Carderi, Joseph, Le Maître and