There was a time period when it could have been said that every portable typewriter introduced to the market was in one or more ways directly responsive to the widely popular, folding CORONA machine.  Many of these machines were extremely light in weight, and many of them also featured a three-bank double-shift keyboard.  Notable machines in this group would have to include the Fox portables (both folding and later non-folding) as well as the Tell/Mitex/Bar-Let, and a number of small thrust-action machines, and even the earliest Underwood (three-bank) portable.  We also can include the NATIONAL and PORTEX in this group.

The machine was certainly invented (or designed, anyway) several years before it was built.  The patents for the machine were first filed in February, 1915, and with subsequent divisions and considerations were not filed until 1920 and 1921.  By this point, the machine was already known as it had entered production in 1917 as the National No. 2. 
The National No. 2 did not have shift keys on both sides, and had no right-side carriage release / margin release lever.  The No. 3 is quite scarce; the No. 5 did have shift keys on both sides (with lock key poking up oddly on the right side only) and had duplicate levers on both carriage ends for carriage release / margin release.  Other than these differences (and some with the paper holder fingers, guide at the print point and some smaller changes) the models were all practically identical.  Some machines of the No. 5 variant, no matter whether National or Portex, are totally nickel-plated but this plating does not always survive well.
The various concerns that built this line of little portables seems confusing as the history of the various Harris / Rex / Demountable corporate entities, which follows since they were built in the same plant.  TYPEWRITER TOPICS of 1923 gives some data, and we have patent information for support.  Apparently, the machine, although surely designed during the First World War, did not appear until about the end of it; the National No. 2 first appeared at the very end of 1917, and supposedly was produced by the National Typewriter Department of Rex Typewriter Company.  In point of fact, the patents are all assigned to Rex itself, in either one of two of that company's successive incarnations (Rex Typewriter Company, incorporated in Illinois, which is the incarnation during ownership and control by American Can Company, and then later Rex Typewriter Company Inc., incorporated in Virginia, the first entity following selloff by American Can and more local ownership, and the time when offices were moved from Chicago back to Fond du Lac -- the break between these two entities being about March 1921.)  The National No. 3 appeared immediately after the No. 2, and it is not known for how long, or if, both models overlapped.  The National No. 5 appeared next, in early 1920, supposedly at first a product of a National Typewriter Company of Fond du Lac.  This seems slightly improbable, but whatever the case we know that the No. 5 eventually also appeared labeled as the Portex No. 5, and that there are examples of National No. 5 and Portex No. 5 machines intermingling in serial numbers in the range 19,000-20,000.  One machine pictued in Darryl Rehr's book ANTIQUE TYPEWRITERS AND OFFICE COLLECTIBLES with a serial above 22,000 also bears the label "Rex Typewriter Corporation" on its front frame, and "Portex" on its paper table.
PORTEX NO. 5  S/N 19882  DAVID B. DAVIS COLLECTION
Why are the serial numbers of National and Portex No. 5 machines mixed?  We don't know, and this makes as much sense as the Harris No. 4 machines having two different serial number blocks (although one example was found recently with a serial in the higher 100,000 block which included its original SEARS ROEBUCK manual, hinting that the high block MIGHT have been for Sears machines only, a hazardous guess.)  Perhaps the National and Portex machines were built concurrently for sale by two different entities, after all; perhaps the supposed "National Typewriter Company" allowed Rex to sell the same machine under license as a condition of the contract.  We may never know at this late date, but this overlap provides yet another interesting question regarding a large line of machines (Harris/Rex/Demountable and National/Portex) that always prove out to be interesting research subjects.
NATIONAL No. 2  /  serial no. 1590

Thomas Fuertig collection


Here we see an early variant of the whole line; serial numbers have been found in the hundreds, and run into the 20,000+ number range for No. 5 machines. 

Note the paper fingers on the carriage, and also note that the shift keys are located on the left side only; a lever-type shift lock is employed. 
EXPRESS  /  no serial number

Thomas Fuertig collection


The National-pattern machine was, at various times through its life, distributed through channels not associated with either the National or Rex operations.  The EXPRESS is one such example, supposedly distributed through an agency in Switzerland.  This machine is actually a sub-variant of the National No. 2 series, and has a paper bail instead of the fingers -- and a "QWERTZ" keyboard arrangement.

Thomas reports that he completely disassembled this machine to search for a serial number, and there absolutely is none.

We are indebted to Thomas Fuertig and Herman Price for the submission of serial number lists that include known machines and information about their markings and thus distribution.
NATIONAL B  / serial 5516

Tilman Elster collection


Here is another machine likely unfamiliar to US collectors; it's a National B, which seems to have been a foreign-sales version of the National No. 3 machine.  Note that this machine differs from the two seen above it by way of having introduced a duplicate set of shift keys on the right side of the keyboard.  There is also a shift lock key on the right side (only) which replaces the locking lever used prior.  This machine has a French "AZERTY" keyboard layout.
NATIONAL No. 3  /  serial 12681

Tilman Elster collection

This is the conventional No. 3 variant of the National, again showing the two sets of shift keys.  The decals change constantly over time in location and style; this variant has an emblem on its paper table, the name NATIONAL on the front, and the model identification spelled out on the front frame.
NATIONAL No. 5  /  serial 20718

Tilman Elster collection

This machine's body is nickel plated, and is extremely attractive.  Note the emblem on the paper table with an eagle; this emblem is also occasionally seen on very late PORTEX machines as well.
PORTEX No. 5  /  serial 19552

Tilman Elster collection

The name PORTEX first appears on these machines at about serial number 16390, and runs occasionally through all of the rest of production along with the name NATIONAL.  We know that both were being built at the Rex Typewriter Company's plant; perhaps the PORTEX was distributed through Rex's channels, and the National through other different channels. 
NATIONAL No. 5  /  serial 22395

Tilman Elster collection

The eagle emblem again is seen on this machine, which has the second-highest known serial number of any machine of this entire line.  Note that it is all-black, and carries no labeling on its front; only on the front frame, and on the paper table.