Reversing the order of up and downsampling
The problem with downsampling is that it can be lossy -- since you're reducing the sampling rate, you can introduce aliasing. So, you can reverse the order whenever downsampling does not result in aliasing.

For example, say your discrete-time signal $x[n]$ contains energy in frequencies up to $f_N/3$, where $f_N$ is the Nyquist frequency. Then, downsampling by a factor of 2 will not affect the signal: the "new" Nyquist frequency is $f_N/2$, so there is no aliasing.

See also Which order to perform downsampling and filtering?.