now I understand,
refurbishment started in 2010 and was completed in 2011 so
-- FROM - TO seem to be the correct version,
-- BETWEEN would mean SOME time between 2010 and 2011 (for example, one month only, not the WHOLE period, which is not the case here),
thank you for explaining,
I wouldn't use either construction. "Between" doesn't work, in my opinion, because there isn't anything between 2010 and 2011; the moment the first has ended, the second has begun. And "from...to" also seems to indicate a similar time span.
Assuming that the refurbishing, as you say, began sometime during the year 2010 and was completed sometime during the year 2011, I'd say, "The building was refurbished in 2010-11."
I'd vote for between. It signifies the entire period of the year in which the refurbishment could have taken place.
"They decorated the living room sometime between March and April" - the idea that this doesn't make sense because there is nothing between them just does not hold up in my version of English, it means from the 1st March until the end of April, a refurbishment took place.