Interpret clustering results after variable transformation
Very interesting!
What you did to your data is simply a feature mapping/transformation. So how this affects the clustering results?

Clustering is not a clearly defined problem but at least we know something about it: It's about internal similarities (patterns) so these similarities should be maintained through the feature transformation. In your example if you found clusters in transformed space, it shows that you have had clusters in original space as well. You just couldn't see them according to the algorithm you used in that space!

For instance if you use Kernelized versions of algorithms you easily find that what they do is nothing but what you did as transformation. They first use a kernel to map the data into new space and then use the algorithm in that space (of course with a bit of theoretical differences/constraints).

To summarize, no transformation produces a fake pattern in the data. In worst case it vanishes the original pattern and in the best case it reveals the pattern which was not visible originally (which is your case).



I mentioned Fake Pattern above so let me say a bit more on it. I think there is a fundamental issue concerning your question: 

You assume that there is a Right clustering that you got after transformation. Actually there is no right clustering! 

We do not have fake pattern! If there is a pattern in a feature space, then that is true! i.e. you found an interesting representation of your data. If it does not match with the labels then either the data is very noisy or wrong features have been chosen to represent classes (maybe more reasons. just these two came to my mind now). If there is no label (your case) be sure there is a correlation between features of those cluster members.