I have created an insight with a date filter, and enabled "Compare the period with", selecting either "Same period previous year" or "Previous period" for all measures (I only have one). In the insight designer, things are labeled as expected: One color with my measure name, and another with measure name - SP year ago.
However, when I get the visualization object for my insight, the SP year ago measure does not have a "title". I can manually compute the title, but is there a way to get it through the API?
Title for PoP/Previous period is not stored in visualization object but instead generated in Analytical Designer from original measure's title, so we can correctly localize it. So to answer your question, there is currently no way to get it from API.
Peter