Among the values for which the dilogarithm and its argument can both be given in closed form are the following four equations:

$Li_2( \frac{3 - \sqrt{5}}{2}) = \frac{\pi^2}{15} - log^2( \frac{1 +\sqrt{5}}{2} )$ (1)

$Li_2( \frac{-1 + \sqrt{5}}{2}) = \frac{\pi^2}{10} - log^2( \frac{1 +\sqrt{5}}{2} )$ (2)

$Li_2( \frac{1 - \sqrt{5}}{2}) = -\frac{\pi^2}{15} - log^2( \frac{1 +\sqrt{5}}{2} )$ (3)

$Li_2( \frac{-1 - \sqrt{5}}{2}) = -\frac{\pi^2}{10} - log^2( \frac{1 +\sqrt{5}}{2} )$ (4)

(from Zagier's *The Remarkable Dilogarithm*)
where the argument of the logarithm on the right hand side is the golden ratio $\phi$. The above equations all have this (loosely speaking) kind of duality, and almost-symmetry that gets broken by the fact that $Li_2(\phi)$ fails to make an appearance. Can anyone explain what is the significance of the fact that $\phi$ appears on the right, but not on the left? Immediately one can see that the arguments on the lefthand side of (2)-(4) are related to $\phi$ as roots of a polynomial, but what other meaning does this structure have?