Looking at the Nuñes couples videos and how they Wind the coil, there is definitely a lot of capacitance between the individual strands. How much actually depends on how close they are to the previous pair. Definitely if you want reproducability and regularity one would have to sort out the relative position of each strand of wire to the other strands. One could number them 1-12 (if using 12) going in a clockwise direction.
Also from balanced transmission line theory and practical work, it is known that the impedance of a twisted pair, depends on the interwinding capacitance and the relative permittivity of the insulation (which is around 3 for most types as far as I can recall). Putting on more turns per length increases the interwinding capacitance and will cause a decrease in measured impedance.
Impedance affects the propagation speed of the electromagnetic wave and therefore it's resonant frequency.
Apart from that there is also a much smaller inter-cable capacitance which should give rise to an additional, but much higher resonance frequency.
Regarding the coil as a transmission line and putting on a sinewave, should at the basic resonant frequency and it's overtones and harmonics, cause standing waves on it, which can cause high voltage (if open of lightly loaded) and no current or high current (if shorted or loaded) with no voltage. Impedance matching is definitely worth exploring - especially in the case of connecting it to a HHO cell with a specific impedance (mainly capacitive).
So to sum up parameters that should theoretically affect reproducability and resulting characteristics:
1. Order the wire strands
2. Connect them in the same order sequence.
3. Use same wire thickness and insulation.
4. Put on the same number of turns per unit of length - or measure the interwinding capacitance to have the same values.
5. Use the same fixture to have the same inter-cable distance and wire length.
Will be Building one soon. Too bad I don't have access to a real Network analyzer (yet..) ;-)