What is the significance of spirituality in The Left Hand Of Darkness?
I think a better question is what is the significance of religion in The Left Hand of Darkness. Spirituality is tied with religion, yes, but spirituality connotes a much more personal and individual belief and practice within a particular religion. I remember the religions of Handara and Yomeshta being more the focus. I think the significance of those two religions is that they closely mirror existing religions on Earth. Handara is similar to an "Eastern" Tao, Zen, or Buddhist religion. Taoism is Chinese and where the standard Yin and Yang come from. Basically, there are two sides to everything and both of those sides are essential to make something stronger, better, smarter, etc. Lightness and darkness are not good and bad, but necessary for harmony and balance.
Yomeshta is more in line with a stereotypical "Western" religion: Judaism, Christianity, etc. The Yomeshta religion stresses that true knowledge comes from the light and darkness brings death.
The long and short of it is that both religions are trying to answer the same questions concerning light and dark, but answering them with completely different methodologies.