Thanks for the tips, man :) I looked into it and made a new gif with 19 colors (basically the same gif with a color palette limited by export options). I also added 2 frames to the cycle :)

You can take a look here : http://pixeljoint.com/pixelart/111929.htm

Looks good!  You may need to check that when you export that there is no compression going on as that can lead to extra colors.  You can probably find out online how to turn the compression off in Photoshop seeing as I'm not familiar with it's ins and outs.

Check this out.   http://yanrishatum.ru/pj/

You can paste the address of your submission and it'll show you how many colors you have, but more importantly for you in this case, it'll show you where the colors are located in your piece just by clicking on the color to the right.  

On this piece I would suggest another frame in the animation.  An extra middle frame could make it a little smoother.  

Havefun!  Welcome.  And as always... keep getting better!