Clarification regarding discrete wavelet transform
Assuming that you have a sufficient number of samples, throwing away odd / even samples does not matter.

The DWT can be thought of as measuring time/frequency content with varying levels of time/frequency resolution.  For non frequency varying signals (like chirps), the odd and even samples both contain the same frequency and time content. (Unless we are talking about some explosive transient that occurs in one sample)  then you would need some a priori knowledge about that.

If the signal is of odd length, taking the odd samples would give you one more sample than the even samples but wouldn't make a difference in terms of information gained.