My first reaction would be that it has something to do with Quantize and possibly the beat grids not being adjusted in Rekordbox, but you said the grids were OK so that is probably not the issue.
Have you tried setting the Hot Cues in Rekordbox? If not, try setting them there and see if they go where you want. It's a lot nicer to set them in Rekordbox because you can zoom into the waveform and line the Cue up visually exactly where you want it.