  We aim to explore the properties of the Be/X-ray binary system MXB 0656-072
from a timing analysis perspective through an investigation of the RXTE/PCA and
Fermi/GBM data during its 2007-2008 type I outbursts. We applied two new
techniques, for the first time, along with the conventional Deeter method to
produce higher-resolution power density spectra (PDS) of the torque
fluctuations. We also investigated the spin frequency evolution of the source
by utilising a pulse timing technique. The PDSs show a red noise pattern, with
a steepness of $\Gamma \sim -2$ and a saturation timescale of $\sim$150 d,
indicating that MXB 0656-072 is a disc-fed source. With the obtained long term
spin frequency evolution, we reveal the torque-luminosity correlation of MXB
0656-072 for the first time. We also demonstrate that the frequency evolution
is largely consistent with the Ghosh-Lamb model. In the RXTE/PCA observations,
the pulsed emission disappears below $\sim$5$\times 10^{35}$ erg s$^{-1}$,
while the profiles remain stable above this value in our analysis time frame.
We show that the magnetic field strength deduced from the torque model is
compatible with the field strength of the pulsar derived from the cyclotron
resonance scattering feature. Utilising the new distance of MXB 0656-072
measured by Gaia, we show that the spectral transition of MXB 0656-072 occurs
at a luminosity that matches the expected theoretical transition from the
subcritical to supercritical accretion regime.
