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An ion of chlorine with a charge of −1 is called a chloride ion/anion.

Are there terms for other ions of chlorine, for example, $\ce{Cl+}$ or $\ce{Cl^2-}?$

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    $\begingroup$ Chloride ion is ok. Everyone will understand chloride anion but that phrase is redundant. Chloride already specifically means the ion $\ce{Cl-}$. // For $\ce{Cl+}$ is use chlorine(I) cation or chlorine(1+) ion. That is such an odd species that I'd explicitly specify the charge and not just say chlorine cation. $\endgroup$ – MaxW Jan 9 at 8:49
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    $\begingroup$ Questions regarding halogen forming cations: chemistry.stackexchange.com/questions/40171/… ... chemistry.stackexchange.com/questions/40643/… $\endgroup$ – Nilay Ghosh Jan 9 at 11:03
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Assuming that "term" in your question refers to chemical nomenclature, the authoritative source of information would be the current edition of IUPAC Red Book [1]. From [1, p. 70]:

IR-5.3.2.2 Monoatomic cations

The name of a monoatomic cation is that of the element with an appropriate charge number appended in parentheses.

[…]

  1. $\ce{I+}$ iodine(1+)

Analogously, the name of $\ce{Cl+}$ is chlorine(1+).

Further from [1, pp. 72–73]:

IR-5.3.3.2 Monoatomic anions

The name of a monoatomic anion is the element name (Table I) modified so as to carry the anion designator ‘ide’, either formed by replacing the ending of the element name (‘en’, ‘ese’, ‘ic’, ‘ine’, ‘ium’, ‘ogen’, ‘on’, ‘orus’, ‘um’, ‘ur’, ‘y’ or ‘ygen’) by ‘ide’ or by directly adding ‘ide’ as an ending to the element name.

Examples:

  1. chlorine, chloride

[…]

Charge numbers and radical dots may be added as appropriate to specify anions fully.

Examples:
14. $\ce{O^2-}$ oxide(2−), or oxide

So, the name for $\ce{Cl^2-}$ is chloride(2−).

Reference

  1. IUPAC. Nomenclature of Inorganic Chemistry, IUPAC Recommendations 2005 (the “Red Book”), 1st ed.; Connelly, N. G., Damhus, T., Hartshorn, R. M., Hutton, A. T., Eds.; RSC Publishing: Cambridge, UK, 2005. ISBN 978-0-85404-438-2.
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    $\begingroup$ @NilayGhosh I was wrong in that discussion, Mithoron was correct. Chloronium is not $\ce{Cl+},$ it is $\ce{R2Cl+}$ (where $\ce{R}$ might as well be $\ce{H}).$ The original paper explains it more clearly IMO—by the way, you could answer that question of yours by simply quoting the onium compounds section:) $\endgroup$ – andselisk Jan 9 at 11:42

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