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We provide several new <i>q</i>-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews&#8217; multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial.