How can I create a “crazy mirror” distorted portrait in Photoshop?

To truly make this effect you need a physical mirror. In the examples you show the faces are seen from multiple angles in the same image. That is not achievable in Photoshop, but we can have some fun with Displacement Maps.

I’m using a portrait by John Torcasio (CC0 Public Domain).

Create a grayscale image with the same dimensions as the original. Paint some smooth blurry shapes. This was made with a gradient with some effects, but you can just paint with a soft brush. Save it as .PSD.

When this image is applied as a displacement map, the original image will be distorted according to the lightness of the displacement map.

Use Filter > Distort > Displace to apply the displacement map. Here you can choose how much the displacement map will affect the image horizontally and vertically. After pressing OK to the settings you will be prompted to select a .PSD file to use as displacement map. Choose the file you just created.

Here are four examples all using that same displacement map, but with different settings.

As you see, the effect can be dramatically different just by using different values. And since you can paint the displacement map anyway you want, the possibilities are endless.

Here the filter is set to Repeat Edge Pixels so the edges of the image might end up looking a little weird. Make sure to use an image which can take a little cropping afterwards.

If the displacement map isn’t smooth enough you will get a completely different and more chaotic result. This can also look cool, but it’s way different than the desired effect.