No. There are many species of pigeons and doves. A little over 300 IIRC, although only one species was domesticated, several thousand years ago. There are several dozen species just in the Genus to which the domestic pigeon, Columba livia, belongs. (The terms "pigeon" and "dove" are effectively interchangeable. The smaller species in the pigeon family are more apt to be called "doves".)
I'll assume when you say pigeons are "one species" you mean to refer to the fact that the domestic pigeon, including it's various and wildly varying breeds; the wild rock pigeon; and the feral pigeon (wild domestic pigeons like you find living in cities, which are somewhat different from, and more various in plumage and behavior than, the pure wild Rock Pigeon) are all the same species. This much (I *think*) is true.
And the funny thing is... no matter how "far out" the breeding went, subsequent generations always returned to the archetype.
Um, no. I don't *think* this is true. First of all feral pigeons never (quite) return to the full form of the true wild Rock Pigeon. The later exists in pure form almost exclusively in remote regions.
I'm almost certain that the "always" is false. I'd bet many of the well established breeds, e.g. carrier pigeons (not to be confused with homing pigeons, but rather the breed with the wattles around its beak and eyes, and the unusual tall and upright stance), breed true to form over however many generations.
Rock Pigeon (This looks like either a wild one or a feral resembling the wild type. Feral pigeons come in many other color patterns as well.) 
Carrier Pigeon 