First the recommendation, and then the question.
I’d be glad to know what you think about Hoppe’s theory.

I can write what I think about your recommendation. If you want to discuss it, summarize the main argument here and we can discuss it. A general referral to read about a topic is not a question and not a basis for a question. That’s homework.
If you just want to give a general reading recommendation, that’s perfectly fine. I’ll consider it. But it doesn’t belong in the responsa section.

I’ll try to explain it as briefly as I can, though I think it’s also worth reading, because I’m really not yet a philosopher who knows how to explain it properly.

Hoppe’s theory in its original form says the following.

Every claim is asserted and decided through argument.
It is impossible to deny claim 1 without entering into an argument and thereby running into a self-contradiction.
Argument is not just sounds in the air, but purposive human action in which a person uses means in order to reach agreement on a claim.
Every argument is a non-violent interaction and an agreement to reach a resolution of disagreement regarding a certain claim.
The validity of the norms underlying argument cannot be denied without running into a self-contradiction, for the same reason as in claim 2.
Those norms are that every person is entitled to control over his body, so that both sides can argue independently, and as a derivative of that, control over the external resources in their possession prior to the argument.
Therefore, any claim against these norms would be self-contradictory, since in argument both sides are seeking a non-violent solution for resolving disagreement.

Again, I emphasize that this theory can be added to in order to make it more precise, and the main point can be approached from several directions.
The point is that there is no logical way to justify a claim that denies the norm of a person’s self-ownership. There will always be a contradiction.
Because of this, I don’t think this is morality in the sense you were talking about (and that doesn’t mean such morality can’t also exist), but it is still something.

Those assumptions themselves are value-laden assumptions, and therefore they too require grounding. And there is no grounding without God. In short, what would you say to me if I don’t want to argue, but rather to kill you and take all your money?

That’s why I think the theory is incomplete. I have various ideas about how to complete it, but I need to work it out in my head and in writing before I discuss it.
I agree that in the final analysis you probably do need divine grounding, but in my opinion that’s true of all reality, not just morality.
Maybe a better way to look at the theory is that it doesn’t come to prove the existence of universal morality, but rather shows what that universal morality ought to be, assuming it exists.

If we all see material reality in the same way, aside from those who, within material reality itself (an important point), see it differently, then when it comes to moral reality, any argument that tries to justify using force against another necessarily cannot be accepted (shown) by everyone, because there will always be that one person against whom the force is being used, who won’t agree.

Considerations of symmetry are irrelevant. When there are moral arguments and they clash, the question of symmetry arises. The question is how there are such arguments at all.

If so, how can one “see” morality?

I didn’t understand the question or how it relates to the discussion.

I believe Hoppe’s theory says what that basic morality ought to be. In your view, what is the way to know what it ought to be? What are the rules? There is a difference between defining morality and arguing for its source, and knowing its content.

The hard question is not what the principles of that basic morality are, but what its validity is. Everyone understands what the basic rules of morality say. There is no significant dispute about that. The question is how one justifies their validity.

I disagree with the claim that everyone understands those same rules. Human beings have different consciences, and I’m not talking here about exceptional cases like psychopaths.
Just one random example out of many: say there is a hungry poor person on a certain street, and next to him there is a bakery owned by a billionaire that gives him nothing (and let’s assume all its employees are also billionaires). One person’s conscience (let’s also say he has no way to donate himself) will lead him to steal or take bread from the bakery by force and give it to the poor person. Another person’s conscience—mine, for example—would not allow that under any circumstances. Am I a psychopath? No. It’s just that my conscience is different. There is much more subjectivity in conscience than in the ordinary senses.

Of course there are disagreements. You can’t deny simple facts. But the disagreements are at the margins.
But all this is irrelevant to the discussion. I explained that the discussion concerns the validity of ethical norms, not their content.