Contact adhesive is used by some but getting it smooth and even can be tricky and you get one shot at having it right.
Mechanical fixing saves the hassle of glues and means it can be replaced easily, but you end up with holes all over it.
Double-sided tape is the best option, but you need one that will stick to it, prepare the surface of the MDF first by roughing it up with a blade like you would a floor before applying tile adhesive then clean the surfaces and stick it on, some for of mechanical fixing is usefull on smaller parts as it stops the HDPE from being moved at all and straining the bond
Aidan