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We show that a new class of helical phase inflation models can be simplyrealized in minimal supergravity, wherein the inflaton is the phase componentof a complex field and its potential admits a deformed helicoid structure. Wefind a new unique complex-valued index $\chi$ that characterizes almost theentire region of the $n_s-r$ plane favored by new Planck observations.Continuously varying the index $\chi$, predictions interpolate fromquadratic/natural inflation parameterized by a phase/axion decay constant toStarobinsky-like inflation parameterized by the $\alpha$-parameter. Wedemonstrate that the simple supergravity construction realizingStarobinsky-like inflation can be obtained from a more microscopic model byintegrating out heavy fields, and that the flat phase direction for slow-rollinflation is protected by a mildly broken global $U(1)$ symmetry. %, which ismildly broken at the inflation energy scale. We study the geometrical origin ofthe index $\chi$, and find that it corresponds to a linear constraint relating\kah moduli. We argue that such a linear constraint is a natural result ofmoduli stabilization in Type \MyRoman{2} orientifold compactifications onCalabi-Yau threefolds with geometric and non-geometric fluxes. Possible choicesfor the index $\chi$ are discrete points on the complex plane that relate tothe distribution of supersymmetric Minkowski vacua on moduli space. Moreprecise observations of the inflationary epoch in the future may provide abetter estimation of the index $\chi$. Since $\chi$ is determined by the fluxesand vacuum expectation values of complex structure moduli, such observationswould characterize the geometry of the internal space as well.

Helical Phase Inflation via Non-Geometric Flux Compactifications: from Natural to Starobinsky-like Inflation