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How dared you speak to me like that?

Is this a correct way to use "dare"? Shouldn't we say?

How dare you speak to me like that?

Dare is sometimes called a semi-modal verb, because it sometimes patterns like a modal, and sometimes like a normal verb.

When it patterns like a modal, it takes inversion, and "not" negation, rather than do-support ("Dare you?" "I dare not").

When it patterns like a normal verb, it takes do-support: ("He didn't dare go", "Do you dare pick it up?")

Both forms are found, and are grammatical.

Personally I am very happy with how dare you? and find how dared you? strange and awkward. But I observe in the iWeb corpus that how dared [pronoun] is slightly more common than how did [pronoun] dare.

Technically speaking, if you were complaining about how someone had spoken to you in the past, you could reasonably use past tense dared.

But idiomatically, the expression How dare you! [do/say something outrageous] is something of a "fixed expression / set phrase", and I suspect some people might have misgivings about modifying dare for tense like that. To my ear, it would be at least slightly more "natural" (though of course it can't be fully natural, given it's riffing of a "frozen form") to use...

How could you dare speak to me like that! (using could as the past tense of can)

But that's a fine point. For most purposes, OP's version (or indeed, How could you have dared...) wouldn't be noticed as either "incorrect" or "unusual".

EDIT: Or perhaps not such a "fine point" after all. Here are some relevant searches in Google Books...

How dare you say that! - 25,100 hits for the "idiomatic standard" present tense version
How could you dare say that! - 211 hits
How could you dare to say that!¹ - 3 hits
How dared you say that! - 3 hits
How did you dare say that! - 0 hits

Note that I added the exclamation marks myself (GB doesn't do punctuation). Obviously, that could be followed by a clause (How dare you say that I'm fat!), but that would be the same for all variations, so the relative preferences should still be valid.

¹ I think it's relevant that another competent native speaker suggested including the infinitive marker to in the past tense version.

The "standard" present tense version is well over 100 times more common than past tense could, but GB has only 6 hits for How dare you to say that! (that's less than 1 in 4000, compared to 1 in 70 for including to in the "forced" past tense alternative).

This suggests to me that although competent native speakers know perfectly well that (for no good reason apart from established idiomatic usage) we don't include to in the standard usage, there's somewhat more uncertainty when it comes to the past tense version. Most likely that's because it's inherently a present tense usage (vociferously objecting to the transgressor, for what he just said/did). We become more uncertain about how to handle things using the past tense simply because idiomatically, such contexts almost never arise anyway.

In short, although this aspect of usage might "intrigue" learners (or native Anglophones with a particular interest in obscure details), the whole issue is more a matter of What would we say if we had to use this expression in a past tense form? (even though in practice we almost never do), rather than What do we say when we use this expression in a past tense form? (answer: We just don't!).

How dared you to speak to me like that?

I have Grammarly installed, and as I type that sentence, it throws an error asking me to change from 'dared' to 'dare!' So yes, dare is common and soothing to our general knowledge of English!

On the other hand, if you change the question into a statement or sentence, you'll understand that it's grammatical:

You dared to speak to me like that

So, it's just past tense.

But 'How dare you...' is a way common in daily English. Check this -

'dare' also serves as an auxiliary verb chiefly used in questions and negatives

"How dare you/he/she/they [do something (present tense)]" is a set expression conveying present anger that an action is being done, has just been done, or was done in the more distant past. The tense of 'dare' does not change, nor does the tense of the verb of the action being complained about.

I come into my room. You have a glass in your hand. How dare you drink my whisky!

You tell me that your brother called me a fool yesterday. How dare he say that!

I recall that a politician, who I don't support, did a bad thing 20 years ago. How dare he do that!

Although the expression is phrased like a question, it is not one. If we wish to know how someone found the courage to do a dangerous thing in the past, we would phrase the question conventionally. We might say "How did you dare to attack the gang of thugs, armed only with a stick?", or "How did he dare to enter the lion's cage, knowing it might kill him?"

To express sorrow or anger that someone behaved badly, we could ask e.g. "How could you speak to my grandmother like that?"

In short, I believe that most native English speakers in that situation would simply not use that phrase, whether through conscious effort or unconscious decisions.

If I was being told a story by my friend, and my friend described another person wronging him/her, I could say "How dare they do that!", but more likely I'd say "I can't believe they'd do that!" or other phrases that aren't frozen into the present like "how dare you".

"How dared they" would certainly be understood, but it's more likely to be distractingly nonstandard. Using "How dared they" seems to fit better into archaic contexts.

How dared you to speak to me like that

is fine. It refers to a past event.

How dare you speak to me like that

is fine. It refers to a current event.

How dared you speak to me like that

is just wrong. It scrambles the time markers in what is an idiomatic construction.

EDIT: Weather Vane and I agree on the substance. And

How did you dare to speak to me like that

seems far more euphonious than "how dared you," but there is a perfectly acceptable past form of the verb "dare."