Solution 1:

I use the following function (added to UINavigationController) to customize the push animation:

- (void) pushController: (UIViewController*) controller
         withTransition: (UIViewAnimationTransition) transition
{
    [UIView beginAnimations:nil context:NULL];
    [self pushViewController:controller animated:NO];
    [UIView setAnimationDuration:.5];
    [UIView setAnimationBeginsFromCurrentState:YES];        
    [UIView setAnimationTransition:transition forView:self.view cache:YES];
    [UIView commitAnimations];
}

I guess you could adapt this code to do whatever animation you want.

Solution 2:

The code which you are looking for:

    [UIView beginAnimations:@"View Flip" context:nil];
[UIView setAnimationDuration:0.80];
[UIView setAnimationCurve:UIViewAnimationCurveEaseInOut];

[UIView setAnimationTransition:
 UIViewAnimationTransitionFlipFromRight
                       forView:self.navigationController.view cache:NO];


[self.navigationController pushViewController:menu animated:YES];
[UIView commitAnimations];

Solution 3:

What you could do is push the next view controller but don't animate it, like so:

[self.navigationController pushViewController:nextController animated:NO];

...and then, in the view controller that is getting pushed in, you could do a custom animation of it's view using CoreAnimation. This might be best done in the viewDidAppear:(BOOL)animated method.

Check out the Core Animation Guide on how to actually do the animation. Look particularly at the implicit animation.

EDIT: updated link