Yeah, there appears to be very little hard research on the subject, but my personal experience from reading neuroscience journals has led me to the understanding that REM sleep removes tiredness. In fact, people with REM sleep behavior disorder -- that is, people who naturally do not go into REM sleep normally, and when they do, they have odd physical behaviors -- complain of chronic tiredness and sometimes end up with serious mental disorders.

The other phases of sleep are even less understood than REM, so I am going to trust my instinct and go with the real research that has been published on polyphasic sleep.