We explore the interplay between symmetry and randomness in quantum information. Adopting a geometric approach, we consider states as (H)-equivalent if related by a symmetry transformation characterized by the group (H). We then introduce the Haar measure on the homogeneous space (\mathbb{U}/H), characterizing true randomness for (H)-equivalent systems. While this mathematical machinery is well-studied by mathematicians, it has seen limited application in quantum information: we believe our work to be the first instance of utilizing homogeneous spaces to characterize symmetry in quantum information. This is followed by a discussion of approximations of true randomness, commencing with (t)-wise independent approximations and defining (t)-designs on (\mathbb{U}/H) and (H)-equivalent states. Transitioning further, we explore pseudorandomness, defining pseudorandom unitaries and states within homogeneous spaces. Finally, as a practical demonstration of our findings, we study the expressibility of quantum machine learning ansatze in homogeneous spaces. Our work provides a fresh perspective on the relationship between randomness and symmetry in the quantum world.