After Brahmopadesham or Upanayana one becomes a Dvija and he is then eligible to recite Vedas, perform Sandhya, perform certain Vedic rituals and Vratas.
Manu Smriti 2.171. They call the teacher (the pupil’s) father because
he gives the Veda; for nobody can perform a (sacred) rite before the
investiture with the girdle of Munga grass.
2.172. (He who has not been initiated) should not pronounce (any) Vedic text excepting (those required for) the performance of funeral
rites, since he is on a level with a Sudra before his birth from the
Veda.
2.173. The (student) who has been initiated must be instructed in the performance of the vows, and gradually learn the Veda, observing the
prescribed rules.
2.176. Every day, having bathed, and being purified, he must offer libations of water to the gods, sages and manes, worship (the images
of) the gods, and place fuel on (the sacred fire).
The verses 173 and 176 are important here. As you can see, one also becomes qualified to perform Deva, Pitri and Rishi Yajnas.
And, according to Vyasa Smriti, one becomes eligible to read the Smritis and the Puranas too.
Brahmanas, Kshatriyas, and Vaishyas are called the twice-born. Their
first births take place when they are delivered of their mother's womb
; their second, when they duly accept the Gayatri Mantra from their
preceptors (21).
Thus made twice-born, and free from all other faults, they become entitled to study the Vedas, Smritis and Puranas (22).
Chapter 1, Verses 21,22
Now, the use of the 24 Mudras during Gayatri Upasana is mentioned in the Devi Bhagavata Purana:
Kalpa is the third (the Vedânga which lays down the ritual and
prescribes rules for ceremonial and sacrificial acts); Nirukta is the
fourth (the Vedânga that contains glossarial explanation of obscure
words, especially those occurring in the Vedas); Jyotish or astronomy
is the fifth; Itahâsa (history) and Purânas is the sixth head; and
Upanisadas is the seventh head. Agni (fire) is the mouth of Gâyatrî;
Rudra is the S’ikhâ (the chief part); Her gotra (lineage) is
Sâmkhyâyana; Visnu is the heart of Gâyatrî and Brahmâ is the armour of
Gâyatrî. Think of this Mahes’varî Gâyatrî in the middle of the Solar
Orb.
Meditating on the Gâyatrî Devî as above, the devotee should shew the following twenty-four Mudrâs (signs by the fingers, etc., in religious
worship) for the satisfaction of the Devî :-- (1) Sanmukh; (2) Sampût;
(3) Vitata (4) Vistrita; (5) Dvîmukha; (6) Trimukha; (7) Chaturmukha;
(8) Panchamukha; (9) Sanmukha; (10) Adhomukha; (11) Vyâpaka; (12)
Anjali; (13) S’akata (14) Yamapâs’a; (15) fingers intertwined end to
end; (16) Vilamba (17) Mustika; (18) Matsya; (19) Kûrma; (20) Varâha;
(21) Simhâkrânta; (22) Mahâkrânta; (23) Mudgara; (24) Pallava.
Book 11, Chapter 16
Any Dvija is eligible to perform Gayatri Upasana and therefore everyone of them can perform these Mudras as well.
