  A combined experimental/theoretical study of the EPR in irradiated
$\beta$-Ga$_2$O$_3$ is presented. Four EPR spectra, two $S=1/2$ and two $S=1$,
are observed after high-energy proton or electron irradiation. One of the S=1/2
spectra (EPR1) can be observed at room temperature and below and is
characterized by the spin Hamiltonian parameters $g_b=2.0313$, $g_c=2.0079$,
$g_{a*}=2.0025$ and a quasi isotropic hyperfine interaction with two equivalent
Ga neighbors of $~\sim$14 G on $^{69}$Ga. The second (EPR2) is observed after
photoexcitation (with threshold 2.8 eV) at low temperature and is characterized
by $g_b=2.0064$, $g_c=2.0464$, $g_{a*}=2.0024$ and a quasi isotropic hyperfine
interaction with two equivalent Ga neighbors of 10 G. A spin $S=1$ spectrum
with a similar g-tensor and a 50\% reduced hyperfine splitting accompanies each
of these, which is indicative of a defect of two weakly coupled $S=1/2$
centers. DFT calculations of the magnetic resonance fingerprint of a wide
variety of native defect models are carried out to identify these EPR centers
in terms of specific defect configurations. The EPR1 center is proposed to
correspond to a complex of two tetrahedral $V_\mathrm{Ga1}$ with an
interstitial Ga in between them. This model was previously shown to have lower
energy than the simple tetrahedral Ga vacancy and has a $2-/3-$ transition
level higher than other $V_\mathrm{Ga}$ related models, which would explain why
the other ones are already in their diamagnetic $3-$ state and are thus not
observed if the Fermi level is pinned approximately at this level. The EPR2
spectra are proposed to correspond to the octahedral $V_\mathrm{Ga2}$. Models
based on self-trapped holes and oxygen interstitials are ruled out because they
would have hyperfine interaction with more than two Ga nuclei and because they
can not support a corresponding $S=1$ center.
