Many species of Tillandsia are ageotropic; they grow irrespective of gravity. You can mount air plants to a support in any position, even upside down. There is no wrong way to mount them.
Tillandsias are hardly ever bothered by insects or mites. They hang in the air with no soil, which is where most pests breed.
The most common disease in air plants is rot resulting from overwatering. Unfortunately rot starts at the base of the plant where water collects and is usually hidden from view. On rare occasions a Tillandsia can be saved by pulling out the rotten leaves and drying the plant off, but usually it is too late. So don’t let it happen!.