Are there chiral compounds that don't rotate plane-polarized light?
Good question.
There's a phenomenon named cryptochirality[1] (meaning “hidden chirality”), when a compound, though chiral, has practically unmeasurable optical rotation activity.

It can happen to molecules with chiral center(s) bearing very similar substituents. (So, no tricks with bonded slightly modified enantiomeric pairs are needed.)

An example is 5-ethyl-5-propylundecane $\ce{CH3[CH2]5-\overset{$*$}{C}(CH2CH3)(CH2CH2CH3)-[CH2]3CH3}$,[2] don't call it “butyl(ethyl)hexyl(propyl)methane”, found e.g. in beans. Its specific rotation is $[\alpha] &lt; 0.001$.

Another, more common example are fats, i.e. triglycerides, $\ce{RCOOCH2\overset{$*$}{C}H(OCOR')CH2OCOR''}$,[1] if containing e.g. only palmitic, oleic and similar long acyls, optical rotation is not demonstrable.[3]

(Related topic is chirality in polymers, see e.g. Q: Chirality on Carbon of PVC molecule.)



References:


Mislow K. &amp; Bickart P.: An Epistemological Note on Chirality. Israel Journal of Chemistry 15, 1–6 (1976)
Wynberg H., Hekkert G.L., Houbiers J.P.M. &amp; Bosch H.W.: The Optical Activity of Butylethylhexylpropylmethane. Journal of the American Chemical Society 87, 2635–2639 (1965)
Schlenk W.: Synthesis and analysis of optically active triglycerides. Journal of the American Oil Chemists’ Society 42, 945–957 (1965)