New graphic evidence (at  http://members.ij.net/rex/swastikanews.html ) shows that the swastika represents overlapping "S" shapes symbolizing "socialism."

The discovery answers the long-standing question of why the swastika was used by the monstrous National Socialist German Workers' Party as its symbol.

Although the swastika was an ancient symbol, one of the reasons that it was chosen and/or maintained by the National Socialist German Workers' Party is because it resembles the "sig rune" (a letter of an ancient Germanic alphabet shown below) which was used as a letter "S."

The double sig rune was also used side-by-side as a symbol of the "SS" division of the National Socialist German Workers' Party and for other words beginning with the letter "S" (see graphic illustrations). The swastika is also a double sig rune, but it is overlapping and not side-by-side.

An internet image search for "double sig rune" or "sig rune" or "sieg rune" or "sowilo" provides more examples.

It started in 1919, when Hitler joined the German Workers' Party, a socialist group. The group sought a new name that would attract socialists in other groups. Other German socialist groups used terms like “National” and “Socialist” in their titles, and the German Workers' Party adopted “National Socialist German Workers’ Party.”

The swastika was chosen with the same goal as the new name, to symbolize socialists joining together as the National Socialist German Workers' Party.

The reason that the old-style swastika was usually turned 45 degrees to the horizontal by the Party was to highlight the letter “S” shape for “Socialism” unified, or "Socialist Solidarity" or for the "Socialist Swastika."

The "S is for Socialism" symbol is a mnemonic device today because a hackneyed abbreviation for "National Socialist German Workers' Party" is used exclusively by media and government schools so that most people who use the abbreviation do not know what the abbreviation abbreviates (National Socialist German Workers' Party).

The images shown (below or at  http://members.ij.net/rex/swastikanews.html) are photos or representations of actual banners and flags used by the National Socialist German Workers' Party.
Two of the flags show the name "National Socialist German Workers' Party" or "National Socialist" incircling or surrounding the swastika. Both swastikas have been turned or face the right to accentuate the "S" shape (turned by the National Socialist German Workers' Party). One of the banners includes an additional word that begins with the letter "S," (Sturmabteilung).
Another banner shows how the National Socialist German Workers' Party used other stylized "S" letters, similar to the stylizing of the swastika. In the banner the swastika itself is overlayed with two additional "S" letters in the common lightning-bolt style of the runes.
Another banner shows a single letter "S" in the lightning-bolt style of the runes, sometimes used for youth organizations.
There is a pennant with heavily stylized "SA" letters, deliberately designed to evoke the overlapping letter "S" shapes of the swastika.

The swastika was used with different meanings long before the National Socialist German Workers' Party. As an even earlier symbol in Sanskrit, the swastika means "good luck," literally "it is good" (Sanskrit is the oldest extant Indo-Aryan language retained in India) and that fit the National Socialist view of merging all socialist groups into one large organization.

In ancient times, the symbol might have also represented the sun or a wheel, thus giving rise to the modern terms "socialist sun" and "wheel of socialism" and the "circle of socialism" for the swastika of socialism.

For more info on the horrid National Socialist German Workers' Party see  http://members.ij.net/rex/swastikamain.html