I have looked at Getting your haircut in Japan already and did not see this word in there.
Usually when going to the barber I simply say:
短{みじか}くしてください
Since my hair is pretty uncomplicated.
Now, I would like to ask them for a "trim" (keep my hair relatively the same length just cleaned up). Would I be able to say the following, or does it sound odd?
整{ととの}えてください
整える should be most suitable. However, just saying 'please put my hair in order' might be a little ambiguous. I'm sure your barber would ask for more details and you would get your desired haircut. But to be sure you might want to mention that you want to keep the length the same.
Therefore you could say:
長{なが}さをそのままで、全体的{ぜんたいてき}に整えてください。
Which pretty much means 'Overall cleanup please, but keep the length as is.'
Another option is to use 揃{そろ}える. (to line up, to make uniform, to put in order, etc.)
I feel this is particularly useful when you have long hair as this 'line up' spirit is quite strong. (At least to me.) But it should make you perfectly understood even when you have short hair.
Personally, I often use 片付{かたづ}ける which literally means 'to tidy up.' Although I believe this is not the first choice of a native Japanese speaker, every hair dresser so far has always understood immediately what I mean.
In case you want any parts trimmed with a machine, this is referred to as 刈{か}り上{あ}げる. Therefore, 刈り上げをお願いします should get you the traditional schoolboy haircut.
