  Higher-order topological phases (HOTPs) hold gapped bulk bands and
topological boundary states localized in boundaries with codimension higher
than one. In this paper, we provide a unified construction and topological
characterization of HOTPs for the full Altland-Zirnbauer tenfold symmetry
classes, based on a method known as nested band inversion surfaces (BISs).
Specifically, HOTPs built on this method are decomposed into a series of
subsystems, and higher-order topological boundary states emerges from the
interplay of their first-order topology. Our analysis begins with a general
discussion of HOTPs in continuous Hamiltonians for each symmetry class, then
moves on to several lattice examples illustrating the topological
characterization based on the nested-BIS method. Despite the example minimal
models possessing several spatial symmetries, our method does not rely on any
spatial symmetry, and can be easily extended into arbitrary orders of topology
in dimensions. Furthermore, we extend our discussion to systems with asymmetric
boundary states induced by two different mechanisms, namely, crossed BISs that
break a $\mathcal{C}_4$ rotation symmetry, and non-Clifford operators that
break certain chiral-mirror symmetries of the minimal models.
