I’ve put a project I call MPFoldTransition up on GitHub. It provides a class you can use to add a fold transition to your application in just a single line of code (in most cases).
What is a fold transition? It’s an animation such as shown above (or as popularized by the Clear todo list app) where an object folds in upon itself until it disappears and the content surrounding it moves in to fill the gap. Except in this case, I’m doing it for an entire UIView or UIViewController and not just a single UITableViewCell. (For an in-depth analysis of what goes into making a folding animation, read my post here.)

Update: The GitHub project now includes flip transitions as well!

Update 2: The folding effect has been improved with gradient shadows.

Update 3: For a touch gesture-enabled container controller with page-flipping (not just a transition), see MPFlipViewController.

Features

There are 3 style bits that can be combined to create 8 different animations.

Direction

Controls whether the current view collapses inward (Fold) or if the next view expands outward (Unfold).

Mode

Determines whether the next view slides in flat (Normal) or rotates in at right angles to the collapsing halves of the previous view (Cubic).

Orientation

Sets whether the fold is vertical or horizontal.

Present a modal view controller

There are extension methods to UIViewController to present or dismiss modal view controllers using fold transitions:

- (void)presentViewController:(UIViewController *)viewControllerToPresent
                    foldStyle:(MPFoldStyle)style 
                   completion:(void (^)(BOOL finished))completion;

- (void)dismissViewControllerWithFoldStyle:(MPFoldStyle)style 
                                completion:(void (^)(BOOL finished))completion;

From your UIViewController subclass you would call this to present your modal view controller:

[self presentViewController:modalViewController
                  foldStyle:MPFoldStyleDefault 
                 completion:nil];

And then call this to dismiss it:

[self dismissViewControllerWithFoldStyle:MPFoldStyleUnfold 
                              completion:nil];

Tip: dismiss your modal controller using a style with the opposite fold bit (Fold or Unfold), so that you get the reverse animation.

Push a view controller onto a navigation stack

There are extension methods to UINavigationController to push or pop a view controller using fold transitions:

- (void)pushViewController:(UIViewController *)viewController
                 foldStyle:(MPFoldStyle)style;

- (UIViewController *)popViewControllerWithFoldStyle:
    (MPFoldStyle)style;

From your UIViewController subclass you would call this to push a new view controller onto the stack:

[self.navigationController pushViewController:detailViewController
                                    foldStyle:MPFoldStyleDefault];

And then call this to pop it back off:

[self.navigationController popViewControllerWithFoldStyle:MPFoldStyleUnfold];

Tip: pop your view controller using a style with the opposite fold bit (Fold or Unfold) from the style used to push it onto the stack, so that you get the reverse animation.

Transition between any 2 views or controllers

MPFoldTransition has class methods for generic view and view controller transitions:

+ (void)transitionFromViewController:(UIViewController *)fromController
                    toViewController:(UIViewController *)toController
                            duration:(NSTimeInterval)duration
                               style:(MPFoldStyle)style
                          completion:(void (^)(BOOL finished))completion;

+ (void)transitionFromView:(UIView *)fromView
                   toView:(UIView *)toView
                 duration:(NSTimeInterval)duration
                    style:(MPFoldStyle)style
         transitionAction:(MPTransitionAction)action
               completion:(void (^)(BOOL finished))completion;

If you really need to get under the hood (e.g. to adjust the timing curve), you can initialize your own instance of MPFoldTransition, set the properties as desired, and then call the perform method to execute the transition.

Storyboard support

You can even incorporate modal or navigation stack fold segues without writing a single line of code! Simply use 1 of the 3 custom UIStoryboardSegue subclasses that are included. These cover modal presentation and push/pop to a navigation stack. Just create a segue between 2 controllers in your storyboard, select Custom as the segue type, then enter MPFoldModalSegue, MPFoldNavPushSegue, or MPFoldNavPopSegue as the Segue Class.