My teacher, although she is clearly a spiritual person, does not actively 'teach' the spiritual side of yoga, but she offers a prayer of gratitude at the end of every session, and invites the class to join her.
I think I only began to find it spiritual when I became more spiritual myself. Now I feel that it is nurturing my body, which is, essentially, the home for my spirit and, therefore, should be cared for.
Yoga also helps you move deeper into a meditative state, connecting your body, mind and spirit, so that when you come to meditate at the end, you may find it easier to connect with the universal spirit and feel alive with it.
That's how it is for me, anyway. I think it's all about how you view it. I see my morning practices as my time with the spirit that surrounds and fills me, my time to feel a connection to everyone else in the world, and with light and goodness and wonder. I burn incense, and try, at the end, to meditate on a spiritual idea or thought. I usually pray afterwards, as I know that prayer (not to God, because I don't believe in God as a higher power, but to the universe - sending intention) is more powerful after meditation, when our minds are in that deeply relaxed state, hence why so many religions practise some form of meditation, although they may not call it that.
When I do yoga and meditation in the morning, it is not only myself who feels more relaxed and at peace, but the energy of the whole house is light and positive. When I don't do it, the energy is low and everyone in the house is irritable. I am constantly astounded at how powerful yoga is as a spiritual 'tool'.