Many familiar varieties, such as Grassmannians, toric varieties, symmetric spaces, and algebraic monoids, arise naturally as representation-theoretic objects. Understanding their geometric and representation-theoretic distinctions and similarities often reveals deeper underlying structures and results.. In this talk, we introduce two new families of spherical varieties: nearly toric and doubly spherical. We will highlight their intriguing connections to combinatorics and representation theory, with a focus on their manifestations among Schubert varieties. This presentation is based on joint work with Nestor Diaz Morera, Pinaki Saha, and Senthamarai Kannan.