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So as far as I've come now in the evening (22:35 local time) is the following:
M has to be 11, 12 and/or 13 since you would have to find a combination working for these numbers.
These numbers you get by the upper and lower bound.
The smallest number you can achive is:
1x10+2x9+3x8+4x7+5x6=110
sqr(110)=10,...
and the biggest number:
10x9+8x7+6x5+4x3+2x1=190
sar(190)=13,...
since only integers are allowed only 11,12,13 work.

Also there are only only some ways to achive these.

let u=uneven e=even
so
uu=u
ue=e
ee=e

So you can only achive u (11 and 13) with
uu ue ue ue ee
And e (12) only with
uu uu ue ee ee or
ue ue ue ue ue

That's as far as I got right now.
I'll post an update if there is any. As you can see the only thing left to do is to find an eample for each one.
I'll keep you updated.

I thought that since I've seen your post two days after you posted it that it'll surely be answered till now, but maybe I'll be the first with the results (If there really are solutions).
Otherwise maybe my thoughts helped somebody