Forgive me for returning to 1.a3 and the Mengarini (1.e4 e5 2.a3) but you were warned in this previous entry.
1.a3 e5 2.e4 Nf6 3.Nc3 Nc6 4.Nf3 g6
This is a reversed version of Glek's Four Knight's line 1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6 4.g3!?.
As you may or may not know the original Halloween Gambit arises after 1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6 4.Nxe5?! which probably is incorrect for several reasons. What is most relevant in our context is the fact that after 4...Nxe5 5.d4, Black's most popular - and probably best - move is 5...Ng6.
Therefore clever players discovered the Reversed Halloween Gambit (1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6 4.g3 (Glek) 4...Nxe4?!) which almost certainly improves over the original. Not only has the knight's best retreat option been eliminated, there is also a couple of kingside weaknesses which at least outweigh White's extra tempo.
The diagram position shows the Double Reversed Halloween gambit which in reality is the original Halloween Gambit with the extra moves a3 and ...g6. Surprisingly it's almost as popular as the more normal looking 5.d4 exd4 6.Nxd4 Bg7 7.Nxc6 bxc6 with a relatively balanced position.
5...Nxe5 6.d4 Nc6
As already mentioned this is a less attractive square than g6.
This is more promising than 7.e5 which was tried in one of the first games with the variation.
Here we see another small point of 1.a3 - the knight cannot go to b4 (as in the parallel line 1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6 4.Nxe5 Nxe5 5.d4 Nc6 6.d5 Nb4). Black's other options are:
a) 7...Ne5 8.f4 Neg4 9.h3 looks promising for White (9.e5 Bc5!? is complicated).
b) 7...Bg7! 8.dxc6 bxc6 is the sensible man's move, transposing to the line 5.d4 exd4 6.Nxd4 Bg7 7.Nxc6 bxc6 mentioned above.
8.e5 Ng8 9.d6 (Dia)
This can be considered the critical position. White obviously has a certain amount of compensation for his piece.
Black opens the centre. G.Jones-Bae, West Bromwich 2005 continued 9...c6 10.Bc4 Qh4 11.Qe2 Bh6 12.g3 Qh5 13.g4 Qh3 14.Bxh6 Nxh6 15.Ne4 Kd8 16.Nf6 b6 17.0–0–0 Qh4 18.Rhg1 Qg5+ 19.Kb1 Ng8 20.f4 Qxf4 21.Rgf1 Qg5 22.Bxf7 Ba6 23.c4 Nxf6 24.exf6 Kc8 25.Bg8 Rxg8 26.f7 Qd8 27.fxg8Q Qxg8 28.Qe7 Bxc4 29.Rf8+ Kb7 30.Qd8 1–0 .
10.exd6 Qf6 11.Nb5 Na6 12.Bc4 Bh6 13.Qe2+ Kf8 14.Be3 Bxe3 15.fxe3 Qh4+ 16.g3 Qh5 17.Qf2 Qf5 18.Qe2 Qh5 19.Qf2 Qf5 20.Qe2 Nh6 21.Rf1 Qh5 22.Qf2 Qf5 23.Qe2 Qe5 24.0–0–0 Kg7 25.Rd5 Qe8 26.Qd2 Rf8 27.Qd4+ f6 28.g4 b6 29.g5 Nf5 30.gxf6+ Rxf6 31.Rdxf5 gxf5 32.Rg1+ Qg6 33.Rxg6+ Kxg6 34.Bd5 Rb8 35.e4 Bb7 36.exf5+ Rxf5 37.Qg4+ Kf6 38.Qh4+ Kg6 39.Bc4 Nc5 40.b4 Ne4 41.Nc7 Nxd6 42.Qg3+ 1–0 Carlsen-Nyysti, Helsinki 2002.
I seem to remember an article about this variation in the Norwegian chess magazine 'Norsk Sjakkblad' by Stokke and Hjortås a few years ago. I will return with the year and issue number and possibly some more analysis whenever I find the relevant issue.
Addendum November 27th
From Stefan Bücker, the editor of Kaissiber, I have received some additional information for those interested in this variation:
The game Carlsen - Nyysti, Helsinki 2002, was published in Kaissiber #20, page 34 (source: Suomen Shakki 2002), in the historical introduction to Maurits Wind's extensive analysis of the Halloween Gambit (pp. 22-51 of that issue). The article starts with Bücker's historical overview, showing that the 4.Nxe5 gambit was invented in 1873 (or earlier) by Dr. Carl Theodor Göring, who is better known as the inventor of the Goering Gambit. Later issues of Kaissiber contained refined analyses on the Halloween Gambit by Maurits Wind.
I can warmly recommend the magazine Kaissiber to anyone who can read German and who is interested in chess history or unorthodox opening theory.