It's even simpler, a lead from pin 4 of the MB88201 to ground initiates it in joystick mode. It can be seen on the picture. In the Neos is one of the three white leads to the contact of the left button. On the 1350  is that loop marked with arrows in the plate. At the side a little cut  of wire, the STROBE signal. And at the underside it lacks a little resistor marked  103.

The 1350 has screened its lead, that fat wire with a ferrite ring.

I'm not sure, only those with that big connector I think.

I've cut the loop and connected the STROBE signal to the left button pin and now it works. May be that selecting the joystick mode is not possible until welding the white leads.

Look at this 1351 mouse, the fat connector, the same case, different sticker and different board inside:

http://www.nightfallcrew.com/gallery/commodore-1351-mouse-for-c64128/

The chip is not a MB88201( I can't see it here but I've seen it on another page).

I bought my 1350 on Ebay named as if it were a 1351 so better is to search for this sticker:

http://i693.photobucket.com/albums/vv298/luki1979/IMG_0787.jpg

Even Apple Macs have a Neos mouse!

http://www.ebay.es/itm/RARE-Vintage-Apple-Macintosh-128k-512k-Mac-Plus-NEOS-Mouse-UNIQUE-/281687620083?pt=LH_DefaultDomain_0&hash=item4195e3f9f3