Honestly, I think everyone lets alignment define way too much about their characters. I'm CE therefore I must murder and steal at every opportunity, I care for no one but myself! I'm lawful good, I must follow every rule and obey every law, I will protect everyone all the time, sacrificing of myself even if it kills me!

Alignments are a guideline, I generally pick it after I decide how I plan to play and not the other way around. A CE character could have a loving family he cares deeply for, and will do anything to keep the safe. He tries to stay out of trouble not because he likes following rules and laws, but because he knows getting arrested will make protecting his family difficult. But he is completely willing to do terrible, awful things, if he can get away with it, or is backed into a corner. Likewise the LG character could be a complete jerk, the kind of person who forces their point of view on everyone around them, for their own good. Someone who hands out justice without considering more then the simple truth, you did something against the rules you must be punished accordingly no matter the circumstances. He protects people, he serves the greater good, but he does not show empathy or compassion on an individual level.

People might argue that those examples are different alignments, but there is a good argument for them being the alignments I assigned them. Anyways, what matters most is how the players plan to play them, so if they can't decide out of character to work together, then I'd suggest to them one should change course.