By their day, these rules of derashah were descriptive only. While Hillel and Shammai may have had the power to make new derashos (there is debate on this point), R’s Aqiva and Yishma’el generation certainly didn’t beyond qal vachomer (deriving from the less obvious case to the more).
Also, none of this necessarily means they invented the rules of derashah or even disagreed over fundamentals. The debate between the two schools of medrash were not over the creation of new laws of derashah. For that matter, it is clear that Hillel’s laws were known to the previous heads of the Sanhedrin, the Benei Beseira. The discussion is over taxonomy; how to understand derashah as being the product of a few clear rules. They could well have simply divided the existing derashos into existing categories, and categorized differently. In fact, we find R’ Yishma’el using ribui umi’ut (a principle of R’ Aqiva’s list) and R’ Aqiva using kelal uperat.
The two series of medrashei halakhah are:
|R’ Aqiva’s school||R’ Yishma’el’s school|
|Shemos||Mekhilta deRabbi Shim’on bar Yochai||Mekhilta (a/k/a Mekhilta deRabbi Yeshima’el).|
|Vayiqra||Sifra (a/k/a Toras Kohanim and Sifra deVei Rav)||Sifrei (lost sometime during the late geonim or early rishonim)|
|Bamidbar||Sifrei Zutah (“Small Sifrei”)||Sifrei (the remaining portion)|
|Devarim||Sifrei||Mekhilta Devarim (largely lost; some portions were recovered from citations including some only found in the Cairo genizah)|
The texts seem to have been redacted in the 3rd and 4th centuries.
The traditional publication of the medrashei halakhah includes four books, mixing the two schools: Mekhilta, Sifra, Sifrei (on Bamidbar) and Sifrei (on Devarim). In fact, the two Sifrei’s often get published as a single volume, despite the difference in style that makes their different origin obvious (once you know to look for it).
A more complete publication would have all seven books, typically published in the order: Mekhilta, Mehilta deR’ Shim’on bar Yochai, Sifra, Sifrei (Bamidbar), Sifrei Zuta, Sifrei (Devarim), Mekhilta Devarim.
The word “mekhilta” is Aramaic, and means “measure” or “rule”. The words “sifra” and “sifrei” are conjugations of the root /spr/, meaning “book” or “writing a book”. Sometimes the word “sifrei” is used to refer to all 4 books.
After Rabbi Yehudah haNasi compiled the Mishnah, organizing halakhah by topic rather than verse, the notion of composing Medrashei Halakhah fell out of use. However, as he was from R’ Aqiva’s school (a student of R’ Aqiva’s student, R’ Meir), that school ended up making greater impact on the final law.