  The Hill operators $Ly=-y"+v(x)y$, considered with complex valued
$\pi$-periodic potentials $v$ and subject to periodic, antiperiodic or Neumann
boundary conditions have discrete spectra. For sufficiently large $n,$ close to
$n^2$ there are two periodic (if $n$ is even) or antiperiodic (if $n$ is odd)
eigenvalues $\lambda_n^-$, $\lambda_n^+$ and one Neumann eigenvalue $\nu_n$. We
study the geometry of "the spectral triangle" with vertices
($\lambda_n^+$,$\lambda_n^-$,$\nu_n$), and show that the rate of decay of
triangle size characterizes the potential smoothness. Moreover, it is proved,
for $v\in L^p ([0,\pi]), \; p>1,$ that the set of periodic (antiperiodic) root
functions contains a Riesz basis if and only if for even $n$ (respectively, odd
$n$) $ \; \sup_{\lambda_n^+\neq
\lambda_n^-}\{|\lambda_n^+-\nu_n|/|\lambda_n^+-\lambda_n^-| \} < \infty. $
