When you cast on your extra stitches, be aware that you need to cast on twice the number of stitches to get the correct length you want in gauge.
The directions say to cast on 136 stitches. What you are doing is casting on the stitches that go up and the stitches that go down. So, if you figure out your gauge, you'll be getting the width that would come from 63 stitches.
So, if you want double the length, cast on 272 stitches.
The person who said to not worry if you're casting on even or odd stitches hasn't taken into account that you're casting on in the middle of a mobius.
The other possibility of construction for making one is to cast on for the width of the scarf in a provisional cast on and knit to the length you want and then kitchner it together. Cast on in a reversible stitch like garter or seed stitch.