We study the formation of intermediate-mass ratio inspirals (IMRIs) triggered by the interactions between two stellar black holes (BHs) and an intermediate-mass BH (IMBH) inhabiting the centre of a dense star cluster. We exploit $N$-body models varying the IMBH mass, the stellar BH mass spectrum, and the star cluster properties. These simulations are coupled with a semi-analytic procedure to characterise the evolution of the remnant IMBH. The IMRIs formation probability attains values $sim 5-50%$, with larger values corresponding to larger IMBH masses. IMRIs map out the stellar BH mass spectrum, thus they might be used to unravel BH populations in star clusters harboring an IMBH. After the IMRI phase, an IMBH initially nearly maximal(almost non-rotating) tends to decrease(increase) its spin. If IMBHs grow mostly via repeated IMRIs, we show that only IMBH seeds sufficiently massive ($M_{rm seed} > 300$ M$_odot$) can grow up to $M_{rm imbh} >10^3$ M$_odot$ in dense globular clusters. Assuming that these seeds form at a redshift $zsim 2-6$, we find that around $1-5%$ of them would reach masses $sim 500-1500$ M$_odot$ at redshift $z=0$ and would exhibit low-spins, $S_{rm imbh} < 0.2$. Measuring the mass and spin of IMBHs involved in IMRIs could help unravelling their formation mechanisms. We show that LISA can detect IMBHs in Milky Way globular clusters with a signal-to-noise ratio SNR$=10-100$, or in the Large Magellanic Cloud with an SNR$=8-40$. We provide the IMRIs merger rate for LIGO ($Gamma_{rm LIG} = 0.003-1.6$ yr$^{-1}$), LISA ($Gamma_{rm LIS} = 0.02-60$ yr$^{-1}$), ET ($Gamma_{rm ET} = 1-600$ yr$^{-1}$), and DECIGO ($Gamma_{rm DEC} = 6-3000$ yr$^{-1}$). Our simulations show that IMRIs mass and spin encode crucial insights on the mechanisms that regulate IMBH formation and that the synergy among different detectors would enable us to fully unveil them. (Abridged)
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