It is well known that cutting a Möbius strip "in half" down the middle results in a band with two twists, homeomorphic to a cylinder. See this question for example.

If instead, one begins the cut one third of the way along the breadth of the Möbius strip, and maintains this distance, the resulting shape is composed of an interlinked Möbius strip and doubly-twisted cylinder. See this question for example.

The familiar argument proceeds by identifying the opposite sides of a square, to obtain a Möbius strip, and trisecting said square.

This explain Möbius strip's dissection into a cylinder and a Möbius strip. It does not explain why the resulting surfaces are interlinked. It is clear that this property is dependent on the ambient space being $\mathbb{R}^3$.

How does one prove that the resulting surfaces are interlinked?

It is well known that cutting a Möbius strip "in half" down the middle results in a band with two twists, homeomorphic to a cylinder. See this question for example.

If instead, one begins the cut one third of the way along the breadth of the Möbius strip, and maintains this distance, the resulting shape is composed of an interlinked Möbius strip and doubly-twisted cylinder. See this question for example.

The familiar argument proceeds by identifying the opposite sides of a square, to obtain a Möbius strip, and trisecting said square.

This explains the Möbius strip's dissection into a cylinder and a Möbius strip. It does not explain why the resulting surfaces are interlinked. It is clear that this property is dependent on the ambient space being $\mathbb{R}^3$.

How does one prove that the resulting surfaces are interlinked?