  A new parametrization of the reionization history is presented to facilitate
robust comparisons between different observations and with theory. The
evolution of the ionization fraction with redshift can be effectively captured
by specifying the midpoint, duration, and asymmetry parameters. Lagrange
interpolating functions are then used to construct analytical curves that
exactly fit corresponding ionization points. The shape parametrizations are
excellent matches to theoretical results from radiation-hydrodynamic
simulations. The comparative differences for reionization observables are:
ionization fraction $|\Delta x_\text{i}| \lesssim 0.03$, 21cm brightness
temperature $|\Delta T_\text{b}| \lesssim 0.7\, \text{mK}$, Thomson optical
depth $|\Delta \tau| \lesssim 0.001$, and patchy kinetic Sunyaev-Zel'dovich
angular power $|\Delta D_\ell | \lesssim 0.1\, \mu\text{K}^2$. This accurate
and flexible approach will allow parameter-space studies and self-consistent
constraints on the reionization history from 21cm, CMB, and high-redshift
galaxies and quasars.
