  We study the spectral diversity of Type Ia supernovae (SNe Ia) at maximum
light using high signal-to-noise spectrophotometry of 173 SNe Ia from the
Nearby Supernova Factory. We decompose the diversity of these spectra into
different extrinsic and intrinsic components, and we construct a nonlinear
parameterization of the intrinsic diversity of SNe Ia that preserves pairings
of "twin" SNe Ia. We call this parameterization the "Twins Embedding". Our
methodology naturally handles highly nonlinear variability in spectra, such as
changes in the photosphere expansion velocity, and uses the full spectrum
rather than being limited to specific spectral line strengths, ratios or
velocities. We find that the time evolution of SNe Ia near maximum light is
remarkably similar, with 84.6% of the variance in common to all SNe Ia. After
correcting for brightness and color, the intrinsic variability of SNe Ia is
mostly restricted to specific spectral lines, and we find intrinsic dispersions
as low as ~0.02 mag between 6600 and 7200 A. With a nonlinear three-dimensional
model plus one dimension for color, we can explain 89.2% of the intrinsic
diversity in our sample of SNe Ia, which includes several different kinds of
"peculiar" SNe Ia. A linear model requires seven dimensions to explain a
comparable fraction of the intrinsic diversity. We show how a wide range of
previously-established indicators of diversity in SNe Ia can be recovered from
the Twins Embedding. In a companion article, we discuss how these results an be
applied to standardization of SNe Ia for cosmology.
