Sacred thread or Yagnobaveetha usually worn over the left shoulder and under the right arms, but on some special occasions, it can/ should be worn around the neck like a garland(Niviti position).
Manu Smriti 2.63. A twice-born man is called upavitin when his right arm is raised (and the sacrificial string or the dress, passed under it, rests on the left shoulder); (when his) left (arm) is raised (and the string, or the dress, passed under it, rests on the right shoulder, he is called) prakinavitin; and nivitin when it hangs down (straight) from the neck.
During rites for the Gods, including Sandhya, Deva Tarpana or Yajna, one should be wearing it in the "Upavita" way.
During rites for the Manes, it should be worn in the "Prachinavita" way (which is just the opposite of Upavita).
And, during the rites to Humans, like Manushya Tarpana etc, it should be hanging like a garland (the "Nivita" way).
Source- A book called "Nitya Karma Puja Paddhati" that i have. Right now i don't remember reading about it in any scriptures so unable to give any direct scriptural references. (I can prove the statement about Prachinavita from scriptures though)
Another reference can be the following passage, which is the footnote on Apasthambha Dharma Sutras 6.18, by the author, Olivile:
The upper garment (or the sacred cord) is worn over the right shoulder and under the left arm (a pattern called pracinavita) at ancestral rites (B 2.10.1), and over the neck like a garland (a pattern called nivita) in rites involving humans, such as sexual intercourse, sacramentary rites, and going to the toilet.
Here is a reference from the Vyasa Smriti:
Then he should sit with his knees flexed and placed on the ground, and his face turned towards the north, catching hold of his holy thread in the posture of a necklace, and offer two libations of water containing barley and sesame unto each spirit of men with the end of a kusha blade held at the root of his little finger. The libations should be cast towards the north.
Vyasa Smriti Chapter 3, Verse 14.
In the verse, Vyasa is describing how to perform the "Manushya Tarpanam". And, in such acts, the thread is to be worn like a garland.
Here is a reference from the Krishna Yajur Veda's 2nd Kanda's 5th Prapathaka.
NivitAm manushyAm prAchinAvitam pitrinAmupavitam devAnAmupa vyayate devalakshanameva tat kurute tithannanvAh tishthan hyAsrutataram vadati tishthannanvAh suvargasya lokasyAbhijithyA Asino yajatyasminneva loke prati yath...||
Sayana says on this Anuvaka- "Upavitam vidhAtum prastauti.." ( oR Here it's describing how the performer of a Yajna should wear the Upavita).
In Manushya rites, Nivita way is prescribed. That's why the wise should perform Manushya rites, as well as rites such as Rishi Tarpana, with sacred thread in the Nivita way. For rites related to the Manes, it should be worn in the PrAchinavita way. For rites related to Gods, including Swadhyaya, it should be worn in the Upavita way (on the left shoulder). In this manner, one becomes endowed with the sign of divinity (devachinhameva kritam bhavati).