  Context. Compact Obscured Nuclei (CONs) are an extreme phase of galaxy
evolution where rapid supermassive black hole growth and$/$or compact
star-forming activity is completely obscured by gas and dust. Aims. We
investigate the properties of CONs in the mid-infrared and explore techniques
aimed at identifying these objects such as through the equivalent width (EW)
ratios of their Polycyclic Aromatic Hydrocarbon (PAH) features. Methods. We
model Spitzer spectra by decomposing the continua into nuclear and star-forming
components from which we then measure the nuclear optical depth, $\tau_N$, of
the $9.8 \mu$m silicate absorption feature. We also use Spitzer spectral maps
to investigate how PAH EW ratios vary with aperture size for objects hosting
CONs. Results. We find that the nuclear optical depth, $\tau_N$, strongly
correlates with the HCN-vib emission line in the millimetre for CONs with a
Pearson correlation coefficient of 0.91. We find the PAH EW ratios technique to
be effective at selecting CONs and robust against highly inclined galaxies
where strong dust lanes may mimic a CON like spectrum by producing a high
$\tau_N$. Our analysis of the Spitzer spectral maps showed that the efficacy of
the PAH EW ratios to isolate CONs is reduced when there is a strong
star-forming component from the host galaxy. In addition, we find that the use
of the inferred nuclear optical depth is a reliable method to identify CONs in
$36^{+8}_{-7}\%$ of ULIRGs and $17^{+3}_{-3}\%$ of LIRGs, consistent with
previous work. Conclusions. We confirm mid-IR spectra to be a powerful
diagnostic of CONs where the increased sensitivity of JWST will allow
identification of CONs at cosmic noon revealing this extreme but hidden phase
of galaxy evolution.
