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.
"Why We Nap" - Stampi - next to impossible to find. It coined the word polyphasic (and monophasic) sleep.
It's exploratory, but pretty good science. It's basically a study of solo sail-boat racers- long distance. They have to get little naps in, but they have to stay awake enough to take advantage of wind and water currents for days and days.
Other than that, at least last time I checked a year or two ago, all sleep scientists had written it off as "sleep deprivation" which they feel they already have a good handle on (it's evil), and therefore they are very disinterested in actually studying it.
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.