The same as other answers, but rendered in terms of origami to define the cut lines.

Label the corners $A,B,C,D$ in rotational order. Then fold and unfold $AB$ onto $DC$ and from this fold identify $A'$ as the midpoint of $BC$, C' as the midpoint of $AD$. Next fold and unfold $BC$ onto $AD$ and from this fold identify $B'$ as the midpoint of $DC$, $D'$ as the midpoint of $AB$.

Now cut along $AA',BB',CC',DD'$ to generate a complete smaller square (originally in the center of the larger square) and eight pieces that are easily paired to get the four remaining smaller squares.

Note that the rotational order may be either clockwise or counterclockwise with equally good validity. Thereby both mirror-image solutions are covered.

The same as other answers, but rendered in terms of origami to define the cut lines.

Label the corners $A,B,C,D$ in rotational order. Then fold and unfold $AB$ onto $DC$ and from this fold identify $A'$ as the midpoint of $BC$, $C'$ as the midpoint of $AD$. Next fold and unfold $BC$ onto $AD$ and from this fold identify $B'$ as the midpoint of $DC$, $D'$ as the midpoint of $AB$.

Now cut along $AA',BB',CC',DD'$ to generate a complete smaller square (originally in the center of the larger square) and eight pieces that are easily paired to get the four remaining smaller squares.

Note that the rotational order may be either clockwise or counterclockwise with equally good validity. Thereby both mirror-image solutions are covered.

The same as other answers, but rendered in terms of origami to define the cut lines.

Label the corners $A,B,C,D$ in rotational order. Then fold and unfold $AB$ onto $DC$ and from this fold identify $A'$ as the midpoint of $BC$, C' as the midpoint of $AD$. Next fold and unfold $BC$ onto $AD$ and from this fold identify $B'$ as the midpoint of $DC$, $D'$ as the midpoint of $AB$.

Now cut along AA',BB',CC',DD'$ to generate a complete smaller square (originally in the center of the larger square) and eight pieces that are easily paired to get the four remaining smaller squares.

Note that the rotational order may be either clockwise or counterclockwise with equally good validity. Thereby both mirror-image solutions are covered.

The same as other answers, but rendered in terms of origami to define the cut lines.

Label the corners $A,B,C,D$ in rotational order. Then fold and unfold $AB$ onto $DC$ and from this fold identify $A'$ as the midpoint of $BC$, C' as the midpoint of $AD$. Next fold and unfold $BC$ onto $AD$ and from this fold identify $B'$ as the midpoint of $DC$, $D'$ as the midpoint of $AB$.

Now cut along $AA',BB',CC',DD'$ to generate a complete smaller square (originally in the center of the larger square) and eight pieces that are easily paired to get the four remaining smaller squares.

Note that the rotational order may be either clockwise or counterclockwise with equally good validity. Thereby both mirror-image solutions are covered.