Having made and designed crystal oscillators for a living for nearly 20 years, I found that you should always use unbuffered inverters as the main oscillator. You might want to add some resistance between pin 2 and the common point of C2 and the crystal.
You might be getting the crystal oscillating in 3rd overtone mode, at around 36 MHz. The propagation delay of the inverters is around 7 ns, which is short enough that you could get 3rd overtone oscillation.
If you have a look at **broken link removed** that has several application examples, mainly there because the unbuffered inverter is unlike most logic gates, in that it can be used as a linear device. The transfer characteristics are shown, and the much lower gain is what is wanted for a crystal oscillator.
Schmitt Trigger is perhaps overstating the difference between a watch crystal, usually at 32.768 kHz, and an AT-cut crystal, which is usually in the 4 - 50 MHz range. Both are quartz resonators, where the oscillation is mechanical. The frequency in both is mainly determined by the mass and stiffness of the quartz. The huge frequency difference between watch crystals and AT-cut crystals cannot be obtained by changing the dimensions alone. A 32.768 kHz AT cut crystal would be about 50 mm thick and would have to be around 1m in diameter, so watch crystals are made by making the quartz into tuning fork shapes, instead of plain disks.
https://en.wikipedia.org/wiki/Crystal_oscillator#mediaviewer/File:Crystal_modes_multilingual.svg
Quartz crystals are made from quartz because it is a piezoelectric material. By simply adding electrodes to the mechanical resonator, the quartz converts the mechanical movement into electricity, (and vice-versa to keep the oscillation going) but the electrical power transfer is a tiny fraction of the mechanical energy, which is why crystals take thousands of cycles to start and stop, and why the electrical circuit can only have a tiny effect on the frequency.