Do not use a mixed grip – one hand pronated, one supinated – while deadlifting. Or at least don't use it for all of your training.
Yes, the mixed grip is much stronger than a regular double-overhand grip because it prevents the bar from rolling in your hands and opening the fingers up. However, this type of grip can lead to some serious imbalances. When pulling this way you'll always have a tendency to twist slightly as you rise up. The side of the supinated hand will tend to rotate forward while the side of the pronated hand will rotate backwards.
While this won't be a problem if you only use the grip once in a while, you risk developing an imbalance that could lead to injuries if you do all of your deadlifts this way. Instead, deadlift with a double overhand grip. That's of course a weaker grip than the mixed one, which isn't necessarily a bad thing since you'll strengthen your grip by training this way. If you continue to have trouble holding onto the barbell, learn the hook grip.
A hook grip involves trapping your thumb between the barbell and your first 2-3 fingers. It will be painful the first few times you try it. But once you master it, it's much stronger than a regular grip and just as strong as the mixed grip, if not stronger.