> how can we get both positive and negative part
Drive the grid positive.
In most all-tube circuits, we don't. It is HARD work. The tube input drops from 1Meg toward 1K. The tube before it won't drive a 1K load well. (When we "must" do it, we use a beefy triode and a step-down transformer.)
But. Here we start with an opamp which can easily drive 1K.
Also the grid input does not drop to 1K as soon as we cross 0.000 Volts. Usually have to go several Tenths Volt positive. We can go 0.1V and still have several K grid impedance. And if the tube gives gain near 10, 0.1V in gives 1V out which is a hot guitar signal.
I don't *know* that it works. Tube-data in this area is scarce. But it seems plausible. Even before considering that this plan has been around and I *think* I have seen reports of it working. (Also not-working when ANYthing is changed recklessly.)
That's for THIS plan. If the grid resistor were 1Meg or more, grid current would shift it, generally negative (for small tubes working easy). Also the cap-coupling will tend to peak-catch large signals and drive the grid negative according to signal level. Both types bias are found in various input stages. Here I think the low 10K resistors are to *avoid* such effects.
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Fuggit. I threw it to the Idiot Assistant (SPICE). Aside from being an idiot, the available tube-models are less trustworthy than two 'same' tubes from different eras/makers. However the numbers gotten agree generally with my expectations, and not very different from the DC number on the published plan (idiot says 9V on plate, plan says 14.3V; I know this model neglects Contact Effect which is small at 300V and large at 25.5V).
I rationalized the +12.5v and -13V opamp supplies to just a 25.5V for the tube alone. This also puts the grid at smack-Zero (actually part-microVolt due to grid current) for easy thinking.
First I swept the grid negative and positive (you can do this with a battery and pot) and watched the plate voltage. This is the basic transfer curve. The on-plan nominal bias is the green dot. You can see there is no sudden change of slope either side. And it does not go all to hell when it crosses 0.0V. Yes, it IS curved, and quickly gets more-curved. But for say the range of 0.4Vpp in 8Vpp out the THD will be far below 10% and pretty 2nd harmonic(*).
We do like to see our waves, not just transfer curves. I hit it with a big sine, 2V peak. Output is pretty bent, but at 12.5V peak output it is "only" 27% THD. The "clean" is still much bigger than the "crap". It's an OK fuzzbox, but many musical styles will want "MORE!!"
(*)This is all for one stage without coupling cap or load. When we line-up two stages and trimmings, bentness will increase. Interestingly it may sorta-cancel for small signals (be sorta-clean), but on large signals I expect it to be "MORE!!" bent. And more 3rd harmonic. Which is more what we want.