Flip between two ViewControllers under the same NavigationController

See this demo for you :

Swift code for flipAnimation :

let  mainStory = UIStoryboard(name: "Main", bundle: nil)
let search = mainStory.instantiateViewControllerWithIdentifier("SecondViewController") as! SecondViewController
UIView.beginAnimations("animation", context: nil)
UIView.setAnimationDuration(1.0)
self.navigationController!.pushViewController(search, animated: false)
UIView.setAnimationTransition(UIViewAnimationTransition.FlipFromLeft, forView: self.navigationController!.view, cache: false)
UIView.commitAnimations()

FlipViewController Animation

Output :

You can add custom transition to the navigation controllers layer just before pushing the view controller.

let transition = CATransition()
transition.duration = 0.3
transition.type = "flip"
transition.subtype = kCATransitionFromLeft
self.navigationController?.view.layer.addAnimation(transition, forKey: kCATransition)
self.navigationController?.pushViewController(viewController!, animated: false)

Note that the animated parameter should be false. Otherwise the default sliding animation will takes place

Swift 5 - I personally like using this approach.

//PUSH
let secondVC = UIStoryboard.init(name: "Main", bundle: Bundle.main).instantiateViewController(withIdentifier: "secondVC") as! SecondVC
self.navigationController?.pushViewController(secondVC, animated: false)
UIView.transition(from: self.view, to: secondVC.view, duration: 0.85, options: [.transitionFlipFromLeft])

//POP
let firstVC = self.navigationController?.viewControllers[(self.navigationController?.viewControllers.count ?? 2) - 2] as? FirstVC
if let firstView = firstVC?.view{
    self.navigationController?.popViewController(animated: false)
    UIView.transition(from: self.view, to: firstView, duration: 0.85, options: [.transitionFlipFromRight])
} else {
    self.navigationController?.popViewController(animated: true)
}