Disclaimer: This is my first time on this sub, and I noticed that answers are held to high standards. If my answer is not suitable, please let me know.

As you have noticed, tigers are often depicted ascending and descending mountains in Chinese art. These poses are associated with certain meanings in terms of feng shui. When tigers are depicted ascending a mountain, they are called 上山虎 shàngshān hǔ (uphill tiger), and when they are depicted descending a mountain, they are called 下山虎 xiàshān hǔ (downhill tiger).

The uphill tiger usually has its head turned so that it is looking back (顏伯龍 (Yan Bolong) 1898-1954, 胡藻斌 (Wu Cho Bun) 1897–1942). The meanings of the pose vary, but it is commonly associated with safety, growth, and auspiciousness because the tiger is ascending the mountain and returning home. It is advised to hang such a painting on an inner-facing wall of your living room.

The downhill tiger usually adopts a fierce pose, crouching and baring its teeth (胡藻斌 (Wu Cho Bun) 1897–1942). This pose has much stronger connotations, since it is believed that tigers will only descend and leave the relative safety of the mountains when they cannot find food. Thus, a downhill tiger is hungry and dangerous. Hence, the ferocity of a tiger in such a pose is said to ward off evil. It is advised to hang such a painting facing the main door of a house to protect the household.

(Downhill Tiger in Traditional Papercutting, Feng Shui of Uphill and Downhill Tigers, Meaning of Uphill and Downhill Tigers).

Edit: Fixed broken link.