Only with such methods it is possible to guarantee adequate mathematical properties for the compression function. Otherwise, if non-invertible functions are composed, the properties of the compound function are unpredictable, especially after the very large number of rounds (i.e. of function compositions) that are used in a practical hash function.
For anyone who might want to design a custom hash function it is critical to understand the importance of invertible functions, which is based on the fact that if you compose two invertible functions you get a function that is also invertible. Only this property allows predictions about the statistical characteristics of the result that is obtained after tens or hundreds of function compositions.
You can take the compression function of e.g. SHA-256, remove the final step that converts the invertible function into a non-invertible function, and you get a decent (invertible) block cipher function.
