  We investigate a family of invertible phases of matter with
higher-dimensional exotic excitations in even spacetime dimensions, which
includes and generalizes the Kitaev's chain in 1+1d. The excitation has
$\mathbb{Z}_2$ higher-form symmetry that mixes with the spacetime Lorentz
symmetry to form a higher group spacetime symmetry. We focus on the invertible
exotic loop topological phase in 3+1d. This invertible phase is protected by
the $\mathbb{Z}_2$ one-form symmetry and the time-reversal symmetry, and has
surface thermal Hall conductance not realized in conventional time-reversal
symmetric ordinary bosonic systems without local fermion particles and the
exotic loops. We describe a UV realization of the invertible exotic loop
topological order using the $SO(3)_-$ gauge theory with unit discrete theta
parameter, which enjoys the same spacetime two-group symmetry. We discuss
several applications including the analogue of "fermionization" for ordinary
bosonic theories with $\mathbb{Z}_2$ non-anomalous internal higher-form
symmetry and time-reversal symmetry.
