(verb A) and (verb B) alternately; now doing A, now doing B
Used in the pattern 〜つ〜つ to express two contrasting or alternating actions happening repeatedly. Attaches to the continuative form (連用形) of verbs. Common in set phrases like 追いつ追われつ (chasing and being chased) and 浮きつ沈みつ (bobbing up and down).
They had a close contest, chasing and being chased by each other.
possessive marker (archaic)
Archaic possessive particle, equivalent to modern の. Survives in place names (e.g., まつ (松) + つ + しま (島) → 松島) and in some compound words. Not used in modern speech.
小舟が波に浮きつ沈みつしていた。
The small boat was bobbing up and down on the waves.
The つ in まつしま (Matsushima) is an old possessive particle, equivalent to modern の.
The two of them talked while going back and forth.
つ is the native Japanese general counter used with kun-readings of numbers (ひとつ, ふたつ), whereas か attaches to on-yomi numbers in Sino-Japanese compounds such as 三か月.
The possessive つ is an Old Japanese particle largely superseded by の. In alternating constructions such as 浮きつ沈みつ, つ derives from the classical perfective auxiliary つ.