  Random matrix theory is a powerful tool for understanding spectral
correlations inherent in quantum chaotic systems. Despite diverse applications
of non-Hermitian random matrix theory, the role of symmetry remains to be fully
established. Here, we comprehensively investigate the impact of symmetry on the
level statistics around the spectral origin -- hard-edge statistics -- and
expand the classification of spectral statistics to encompass all the 38
symmetry classes of non-Hermitian random matrices. Within this classification,
we discern 28 symmetry classes characterized by distinct hard-edge statistics
from the level statistics in the bulk of spectra, which are further categorized
into two groups, namely the Altland-Zirnbauer$_0$ classification and beyond. We
introduce and elucidate quantitative measures capturing the universal hard-edge
statistics for all the symmetry classes. Furthermore, through extensive
numerical calculations, we study various open quantum systems in different
symmetry classes, including quadratic and many-body Lindbladians, as well as
non-Hermitian Hamiltonians. We show that these systems manifest the same
hard-edge statistics as random matrices and that their ensemble-average
spectral distributions around the origin exhibit emergent symmetry conforming
to the random-matrix behavior. Our results establish a comprehensive
understanding of non-Hermitian random matrix theory and are useful in detecting
quantum chaos or its absence in open quantum systems.
