IIRC, when you move the shifter forward to the park position, the linkage on the tranny end is supposed to move counter clockwise. What you're describing, means that when the transmission is in P, the shifter is in the L position.
I think that has to do with the orientation of the bar that connects the cable to the tranny. If when facing the engine, it's on your left, then you should be okay. See if you can unbolt it and flip it to the other side (you may have to adjust the bracket that holds the cable), or maybe use back the old linkage and bracket.
BTW, save your neutral start switch and speed sensor from the old tranny.