How is it possible to have these reversed hashes available on the web?

When we say that a hash function h is a one-way function, we mean that

• given some fixed string w, it's "easy" to compute h(w), but
• given f(x) for some randomly-chosen string x, it's "hard" to find a string w where f(w) = f(x).

So in that sense, if you have a hash of a string that you know literally nothing about, there is no easy way to invert that hash.

However, this doesn't mean that, once you hash something, it can never be reversed. For example, suppose I know that you're hashing either the string YES or the string NO. I could then, in advance, precompute h(YES) and h(NO), write the values down, and then compare your hashed string against the two hashed values to figure out which string you hashed. Similarly, if I knew you were hashing a number between 0 and 999,999,999, I could hash all those values, store the results, then compare the hash of your number against my precomputed hashes and see which one you hashed.

To directly answer your question - the sites that offer tables of reversed hashes don't compute those tables by reversing the hash function, but rather by hashing lots and lots and lots of strings and writing down the results. They might hash strings they expect people to use (for example, the most common weak web passwords), or they may pick random short strings to cover all possible simple strings (along the lines of the number hashing example from above).

Since cryptographic hash functions like SHA1, SHA2, SHA2, Blake2, etc., are candidates to one-way functions there is no way to reverse the hashing.

So how do they achieve this; they may choose three ways;

1. Build a pair database (x, hash(x)) by generating the hash of the well-knowns string; the cracked password list, the English dictionary, Wikipedia text on all languages, and all strings up to some bound like 8;

   This method has a huge problem, the space to store all pairs of input and their hash.

2. Build a rainbow table. Rainbow table is a time-vs-memory trade. Before starting to build the select table parameters in order to cover the target search space.

   See Rainbow crack for details of password cracking.

3. Combine both. Due to the target search space, not all well-known strings, passwords, etc. can be placed in the Rainbow table. For those, use the 1. option.

Don't forget that some of them also providing online hashing tools. Once you asked to hash a value, it is going to enter their database/rainbow table, and when you later visit the site and asked the pre-image of the hash that you have stored, surprise they have it now! If the text is sensitive don't use online hashing services.

There is no process for reverse hashing. You just guess a password and hash it. You can make big databases of these guesses and hashes for reverse lookup, but it's not reversing the hash itself. Search for "rainbow tables" for more details.

Those website does not preform any kind of reverse hashing. There are tables called "Rainbow tables". Those rainbow tables are precomputed table for caching the output of cryptographic hash functions. They got lots and lots of strings and calculated hash values for them and when someone search a hash value they lookup the corresponding value from table and display is.