Could the Enigma algorithm be classified as a Feistel network? The Enigma algorithm is a encryption method that was developed (I believe) by the Germans in WWII. It went a little something like this:

When a letter was typed on the keyboard of the Enigma machine, it was first sent through the first rotor, which would shift the letter according to its present setting. The new letter would then pass through the second rotor, where it would be replaced by a substitution according to the present setting of the second rotor. This new letter would in turn pass through the third rotor, again being substituted accordingly. Next, this new letter would be bounced off of a reflector, and back through the three rotors in reverse order. The trick that made Enigma so powerful for its time though, was the spinning of the rotors. As the plain text letter passed through the first rotor, the first rotor would rotate one position. The other two rotors would remain stationary until the first rotor had rotated 26 times (the number of letters in the alphabet and therefore, one full rotation). Then the second rotor would rotate one position. After the second rotor had rotated 26 times (26X26 letters, since the first rotor has to rotate 26 times for every time the second rotor rotates), the third rotor would rotate one position. The cycle would continue like this for the entire length of the message. The result was a shifting shift. In other words, an s could be encoded as a b in the first part of the message, and then as an m later in the message. This principle of the shifting rotors allowed for 26X26X26 = 17576 possible positions of the rotors.
In order for the recipient to decode the message, they would need to know the initial settings of the rotors, and then put the cipher text through the machine to find the plain text. The Germans devised a system by which all of the recipients would set their rotors to predetermined settings according to the date. Each clerk had a book detailing the settings for each day.

Although (I believe) a true Feistel involves splitting into two halves (which doesn't happen here) and where one old half == other new half after a round of encryption, could this algorithm be classified as a Feistel network?
Enigma is not a Feistel cipher. A "Feistel cipher" is a block cipher with a specific structure, namely the whole business with the two halves, the combination of one half with a (one-way) function of the other half and a reversible operation (e.g. XOR), and the swap. See the Wikipedia page which has nice schematics.

So considering Enigma as a kind of Feistel cipher is like considering a salmon as a kind of chicken: yeah, it has no beak, no feathers, no legs and no wings, but could it be a called a chicken ? Strangely enough, I tend to answer no to that. Even though both the salmon and the chicken are edible. But being edible is not sufficient to grant chickenness. Similarly, being a symmetric cipher is not sufficient to grant Feistelness.