  We show that the cross sections for a broad range of resonant {\it inelastic}
processes accompanied by excitation exchange (such as spin-exchange, F\"orster
resonant, or angular momentum exchange) exhibit an unconventional
near-threshold scaling $E^{\Delta m_{12}}$, where $E$ is the collision energy,
$\Delta m_{12}=m_1'+m_2'-m_1-m_2$, and $m_i$ and $m_i'$ are the initial and
final angular momentum projections of the colliding species ($i=1,\,2$). In
particular, the inelastic cross sections for $\Delta m_{12}=0$ transitions
display an unconventional $E^0$ scaling similar to that of elastic cross
sections, and their rates vanish as $T^{\Delta m_{12}+1/2}$. For collisions
dominated by even partial waves (such as those of identical bosons in the same
internal state) the scaling is modified to $\sigma_\text{inel}\propto E^{\Delta
m_{12} +1} $ if $\Delta m_{12}$ is odd.
  We present accurate quantum scattering calculations that illustrate these
modified threshold laws for resonant spin exchange in ultracold Rb+Rb and
O$_2$+O$_2$ collisions. Our results illustrate that the threshold scaling of
collision cross sections is determined only by the energetics of the underlying
process (resonant vs. exothermic) rather than by whether the internal states of
colliding particles is changed in the collision.
