We show that asymmetries in total intensity and linear polarization between the radio jets and counter-jets in two lobed Fanaroff-Riley Class I ( FR I ) radio galaxies , B2 0206+35 ( UGC 1651 ) and B2 0755+37 ( NGC 2484 ) , can be accounted for if these jets are intrinsically symmetrical , with decelerating relativistic outflows surrounded by mildly relativistic backflows . Our interpretation is motivated by sensitive , well-resolved Very Large Array imaging which shows that both jets in both sources have a two-component structure transverse to their axes . Close to the jet axis , a centrally-darkened counter-jet lies opposite a centrally-brightened jet , but both are surrounded by broader collimated emission that is brighter on the counter-jet side . We have adapted our previous models of FR I jets as relativistic outflows to include an added component of symmetric backflow . We find that the observed radio emission , after subtracting contributions from the extended lobes , is well described by models in which decelerating outflows with parameters similar to those derived for jets in plumed FR I sources are surrounded by backflows containing predominantly toroidal magnetic fields . These return to within a few kpc of the galaxies with velocities \approx 0.25 c and radiate with a synchrotron spectral index \alpha \approx 0.55 . We discuss whether such backflow is to be expected in lobed FR I sources and suggest ways in which our hypothesis can be tested by further observations .