The best way to get the PS3 controller back together is to leave the L1/R1 buttons out of it until ya have the rest of it together, they just snap right in after the controller is together, makes messing with those center pieces easier. On the latest version controller they redesigned it and did away with them.
The wiring has to be crossed (reversed) to work, but they don't have to physically cross to pull that off, outside lead to inside trace, inside lead to outside trace and so on, so in that pic it's..
X-Axis (Do only for Legacy)
Light Green - LS Outside Lead to RS Inside Trace
Green - RS Inside Lead to LS Outside Trace
Violet - RS Outside Lead to LS Inside Trace
Pink - LS Inside Lead to RS Outside Trace
Y-Axis (Do only for Legacy Southpaw)
Light Blue - LS Inside Lead to RS Outside Trace
Blue - RS Outside Lead to LS Inside Trace
Magenta - LS Outside Lead to RS Inside Trace
Red - RS Inside Lead to LS Outside Trace
Do BOTH X-Axis and Y-Axis for SouthpawThe Gray/White are for the Sticks buttons and it really depends on which version of controller ya have as to the best way to go about swapping them around, but traces do have to be cut for that to get them swapped.