  We observed $\lambda$3mm \cotw, \coth, \coei, \hcop, HCN and CS emission from
diffuse molecular gas along sightlines with \EBV\ $\approx$ 0.1 - 1 mag.
Directions were mostly chosen for their proximity to sightlines toward
background mm-wave continuum sources studied in \hcop\ absorption, at positions
where maps of \cotw\ at 1\arcmin\ resolution showed surprisingly bright
integrated CO J=1-0 emission \WCO\ = 5-12 K-\kms\ but we also observed in L121
near \zoph. Coherence emerges when data are considered over a broad range of
\cotw\ and \coth\ brightness. \WCO/\Wth\ and N(\cotw)/N(\coth) are 20-40 for
\WCO\ $\la 5$ K-\kms\ and N(CO) $\la 5\times 10^{15}\pcc$, increasing with much
scatter for larger \WCO\ or N(CO). N(\coth)/N(\coei) $> 20-40$ ($3\sigma$) vs.
an intrinsic ratio $^{13}$C/$^{18}$O = 8.4, from a combination of selective
photodissociation and enhancement of \coth. The observations are understandable
if \cotw\ forms from the thermal recombination of \hcop\ with electrons, after
which the observed \coth\ forms via endothermic carbon isotope exchange with
$^{13}$C\p. \WCS/\WCO\ increases abruptly for \WCO\ $\ga 10$ K-\kms\ and
\WCS/\Whcop\ is bimodal, showing two branches having N(CS)/N(\hcop) $\approx 5$
and 1.25. Because CO formation and \hcop\ excitation both involve collisions
between \hcop\ and ambient electrons, comparison of the CO and \hcop\ emission
shows that the CO hotspots are small regions of enhanced N(CO) occupying only a
small fraction of the column density of the medium in which they are embedded.
\hcop/CO and HCN/CO brightness ratios are 1-2\% with obvious implications for
determinations of the true dense gas fraction.
