The tristropha [in the introit psalm verses for modes I, III, and VII] should be sung with a “three-fold rapid percussion of the note,” or with a “three-fold rapid blow, like someone knocking with his hand.” (Aurelian of Réomé, mid 9th cent.)

Every neume is formed of the two motions, upwards [acute accent] and downwards [grave accent], except the repercussed and simple neumes. (Guido of Arezzo, early 11th cent.)

Guido’s commentator describes neumes which “are double or triple in the repercussion of the same sound.” (Aribo, late 11th cent.)

We call a virgula or punctum a simple neume; a repercussed neume is one that Berno calls distropha or tristropha. (John Cotton, early 12th cent.?, quoting Berno of Reichenau, early 11th cent.)

Most of the translations above are from Gregorian Chant according to the Manuscripts by Dom Gregory Murray, p. 40.

Formerly, the individual elements of this figure [i.e., the strophicus] were characterized by an impulse of the voice. . . .  It would be best to repercuss the apostropha gently and softly. (Dom André Mocquereau, early 20th cent.)  Note that this quote is taken from the introduction to the Liber Usualis, where the words referring to the apostropha (apostropham) were rendered in English as “at the beginning of each distropha or tristropha.”  Nearly the same Latin wording was retained in Dom Gajard's revision in the Antiphonale Monasticum.

From the primitive Gregorian notation it is evident that two or more notes on the same pitch and the same syllable are never to be conjoined into one sound: hence each of the notes in the strophicus, trigon, and every other grouping of this sort is to be repercussed.  In passing from one word to another on the same vowel and on the same pitch, a repercussion is made. (Liber Hymnarius, 1983; tr. Peter Jeffery)

The same quotes in Latin:

Terna gratulabitur vocis percussione . . . trinum, ad instar manus verberantis, facias celerem ictum. (Aurelianus Reomensis)

Motus vocum . . . fit arsi et thesi, id est, elevatione et depositione: quorum gemino motu, id est arsis et thesis, omnis neuma formatur, preter repercusse aut simplices. (Guido d’Arezzo)

. . . cum duplices aut triplices in eiusdem sunt soni repercussione. (Aribo Scholasticus)

Simplicem autem neumam dicimus virgulam vel punctum: repercussam vero, quam Berno distropham vel tristropham vocat. (Johannes Affligemensis)

Singula hujus figurae elementa olim nonnulla vocis reparatione discernebantur. . . .  Optimum quidem esset apostropham leniter molliterque repercutere. (Andreas Mocquereau)

E notatione gregoriana primitiva constat duas aut plures notulas eiusdem gradus in eadem syllaba numquam coniugi in unum sonum: unde unaquæque notularum in strophicis, in trigonis aut in omni alio huiusmodi concursu, repercutiatur.  Cum a verbo ad alium in eadem vocali transitur, si in eodem gradu, fit repercussio. (Liber Hymnarius)

These repercussions are more analogous to what is known in modern music as portato, mezzo-staccato, or articulated legato than to pure staccato.  Musicians sometimes need to be reminded that staccato itself simply means detached, not as short as possible (staccatissimo), and not accented.  It must be kept in mind that the repercussions in chant occur under the umbrella of an overarching legato line.  They should be executed tastefully and gracefully, without exaggeration, with only the amount of detachment necessary for the notes to be heard distinctly, which is relative to the acoustics of the room.

I have prepared two PDF handouts on this topic:

Rearticulation and Upper Auxiliary Notes

Repercussion (Rearticulation) according to the Solesmes Method