cycleway=opposite does not show an opposite cycleway, but a road that one is allowed to go ‘against the one way’ by bicycle, without a separate cycleway for the purpose. If there is a separate opposite-direction cycleway, one should use cycleway=opposite_track or draw a separate way. If there is a single way, but with a separate lane for cyclists going the other direction, one can use cycleway=opposite_lane.
These are by the way all rendered on Osmarender: If cyclists are allowed against the one-way traffic, the arrow for the one-way traffic is solid green, if they are not it is white with a green border. It also shows cycleway=lane and cycleway=track (as well as opposite_lane and opposite_track) by a green border on the road. Unfortunately the same does not hold for the cyclemap, which does recognize cycleway=lane, but not cycleway=track.