How to Recover Data when OSX cannot?
You can't recover data directly - in the sense of recovering data lost by accidentally deleting files or formatting a volume - in Internet Recovery Mode/Recovery Mode because no appropriate tools are available in the Base OS X system.

You may clone a disk or volume with dd to an external drive and check the data on a second Mac using a recovery app like Data Rescue/Disk Warrior or Lazesoft Mac Data though. IIRC some of the apps even allow you to create a recovery DVD/thumb drive and by booting from this modified recovery OS X you may check your Mac's disk directly.