In answer to your question, yes you can still check in but read below for all the hassles associated with this.

I just did the same thing on a CX flight, and called their Internet center, and they indicated that they would put a note in the flight record indicating that the name had been switched around. They also indicated that because my name was backwards, I would have to keep all my stubs and then file for mileage credit. They also said that priority would be affected in case of op-up since my FF number is listed as "does not match". I ended up paying to correct the entry rather than go through all this hassle.

Penlin