  We characterize phases of the compass ladder model by using degenerate
perturbation theory, symmetry fractionalization, and numerical techniques.
Through degenerate perturbation theory we obtain an effective Hamiltonian for
each phase of the model, and show that a cluster model and the Ising model
encapsulate the nature of all phases. In particular, the cluster phase has a
symmetry-protected topological order, protected by a specific
$\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry, and the Ising phase has a
$\mathbb{Z}_2$-symmetry-breaking order characterized by a local order parameter
expressed by the magnetization exponent $0.12\pm0.01$. The symmetry-protected
topological phases inherit all properties of the cluster phases, although we
show analytically and numerically that they belong to different classes. In
addition, we study the one-dimensional quantum compass model, which naturally
emerges from the compass ladder, and show that a partial symmetry breaking
occurs upon quantum phase transition. We numerically demonstrate that a local
order parameter accurately determines the quantum critical point and its
corresponding universality class.
