  We show that the geometry of the asymptotic infinities of Minkowski spacetime
(in $d+1$ dimensions) is captured by homogeneous spaces of the Poincar\'e
group: the blow-ups of spatial (Spi) and timelike (Ti) infinities in the sense
of Ashtekar--Hansen and a novel space Ni fibering over $\mathscr{I}$. We embed
these spaces \`a la Penrose--Rindler into a pseudo-euclidean space of signature
$(d+1,2)$ as orbits of the same Poincar\'e subgroup of O$(d+1,2)$. We describe
the corresponding Klein pairs and determine their Poincar\'e-invariant
structures: a carrollian structure on Ti, a pseudo-carrollian structure on Spi
and a "doubly-carrollian" structure on Ni. We give additional geometric
characterisations of these spaces as grassmannians of affine hyperplanes in
Minkowski spacetime: Spi is the (double cover of the) grassmannian of affine
lorentzian hyperplanes; Ti is the grassmannian of affine spacelike hyperplanes
and Ni fibers over the grassmannian of affine null planes, which is
$\mathscr{I}$. We exhibit Ni as the fibred product of $\mathscr{I}$ and the
lightcone over the celestial sphere. We also show that Ni is the total space of
the bundle of scales of the conformal carrollian structure on $\mathscr{I}$ and
show that the symmetry algebra of its doubly-carrollian structure is isomorphic
to the symmetry algebra of the conformal carrollian structure on $\mathscr{I}$;
that is, the BMS algebra. We show how to reconstruct Minkowski spacetime from
any of its asymptotic geometries, by establishing that points in Minkowski
spacetime parametrise certain lightcone cuts in the asymptotic geometries. We
include an appendix comparing with (A)dS and observe that the de Sitter groups
have no homogeneous spaces which could play the r\^ole that the celestial
sphere plays in flat space holography.
