Communion in more than one kind is a relatively recent introduction. The Roman Missal of 1962 does not generally permit it, nor does the Interim Missal published in 1965. Both of those publications have the priest saying only "Corpus Christi" or "The Body of Christ" respectively¹.
As you say, there is nothing in the recent editions of the General Instruction of the Roman Missal to indicate that the order is important. In cases like this, it's useful to go back to the last concrete instruction, which appears to have been given in 1969 in the Institutio Generalis Missalis Romani in the first post-Consiliar Missal.
This is what it has to say on the matter, with my translation (which is of course open to correction).
De Communione sub utraque specie
Ritus Communionis sub utraque specie quando Communionem
suscipientes directo e calice bibunt.
- Si adest diaconus vel alius sacerdos assistens:
c) Singuli communicantes accedunt, debitam reverentiam faciunt, et stant coram celebrante. Celebrans hostiam elevando dicit: Corpus Christi, communicandus autem respondet: Amen, et a celebrante Corpus Christi accipit.
d) Communicandus deinde transit ad diaconum et stat coram eo. Diaconus dicit: Sanguis Christi, communicandus respondet: Amen et diaconus porrigit ei purificatorium et calicem, quem communicandus ipse, pro opportunitate, manibus suis ori admovet. Communicandus, tenens manu sinistra purificatorium sub ore, attendens ne quid Sanguinis defluat, paulum e calice bibit, et postea recedit; diaconus autem partem externam calicis purificatorio absterget.
Of Communion in both kinds
The Rite of Communion in both kinds when the recipients of Communion drink directly from the chalice.
- If there is a deacon or another assistant priest:
c) The communicants approach in single file, make a reverence and stand before the celebrant. The celebrant raises the host, saying "The Body of Christ"; the communicant responds: Amen and receives the Body of Christ from the celebrant.
d) The communicant then passes to the deacon, standing before him. The deacon says: "The Blood of Christ"; the communicant responds, Amen, and the deacon extends to him the purificator and the chalice, which the communicant himself raises, if appropriate, to his mouth. The communicant, holding the purificator in his left hand under the mouth, taking care so that nothing of the Blood may drip, drinks a little from the chalice, and afterwards recedes; the deacon wipes the outer part of the chalice with the purificator.
There are specific instructions laying down the order in which the Sacrament is received, and even how it is done.
It's interesting to note in passing that this Missal does not envisage Extraordinary Ministers of Holy Communion: only priests and deacons distribute the sacrament; and that kneeling to receive is at least discouraged (it's never been forbidden and is now explicitly allowed). It's also not envisaged here that the faithful take the chalice themselves, and that's now explicitly forbidden.
¹ The 1965 form in England & Wales was "The Body of Christ". The form for use in Scotland, which has always been more Presbyterian, was still in Latin.