The quintics are the lowest–order planar Pythagorean–hodograph (PH) curves suitable for free–form design, since they can exhibit inflections. A quintic PH curve r(t) may be constructed from a complex quadratic pre–image polynomial w(t) by integration of r′(t)=w2(t), and it thus incorporates (modulo translations) six real parameters — the real and imaginary parts of the coefficients of w(t). Within this 6–dimensional space of planar PH quintics, a 5–dimensional hypersurface separates the inflectional and non–inflectional curves. Points of the hypersurface identify exceptional curves that possess a tangent–continuous point of infinite curvature, corresponding to the fact that the parabolic locus specified by the quadratic pre–image polynomial w(t) passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on r(t) are incurred by a close proximity of w(t) to the origin of the complex plane. These observations provide useful insight into the disparate shapes of the four distinct PH quintic solutions to the first–order Hermite interpolation problem. Left: The four canonical–form quintic PH curves r(t) determined by the two end derivatives di=2.0+1.5i, df=1.5–4.5i. Right: the corresponding pre–image polynomials w(t), showing the footpoints of the origin on them. •The presence or absence of inflections on planar quintic PH curves is characterized.•The space of planar PH quintics is partitioned by curves that possess points of infinite curvature.•Proximity of the pre-image parabola to the origin of the hodograph plane incurs tight loops.•These observations provide insight into the behavior of planar PH quintic Hermite interpolants.