It is important to treat reversed hide leather in the same way as the rest of the saddle in terms of cleaning and conditioning, with the one difference that you stick to using glycerine saddle soap and a thicker conditioner occasionally (such as Effax or Aussie Leather Care), just wiping over lightly rather than using oil. The oil is likely to penetrate too much and could affect the foam inside.
Do not worry that the 'fluffy' texture of the reversed hide disappears once it has been conditioned; this is how it should be.