Given Hilbert space operators A,B∈B(H), let δA,B(X)=(LA−RB)(X)=AX−XB and △A,B(X)=LARB(X)=AXB for all X∈B(H). The pair (A,B) satisfies (the Putnam–Fuglede) commutativity property δ, respectively △, if δAB−1(0)⊆δA⁎B⁎−1(0), respectively (△AB−1)−1(0)⊆(△A⁎B⁎−1)−1(0). Normaloid operators do not satisfy either of the properties δ or △. This paper considers commutativity properties (δA,λB)−1(0)⊆(δA⁎,λ¯B⁎)−1(0) and (△A,B−λ)−1(0)⊆(△A⁎,B⁎−λ¯)−1(0) for some choices of scalars λ and normaloid operators A, B. Starting with normaloid A,B∈B(H) such that the isolated points of their spectrum are normal eigenvalues of the operator, we prove that: (a) if (0≠)λ∈isoσ(LARB) then (△A,B−λ)−1(0)⊆(△A⁎,B⁎−λ¯)−1(0); (b) if 0∉σp(A)∩σp(B⁎) and 0∈isoσ(LA−RλB) then (δA,λB)−1(0)⊆(δA⁎,λ¯B⁎)−1(0). Let σπ(T) denote the peripheral spectrum of the operator T. If A, B are normaloid, then: (i) either dim(B(H)/(△A,B−λ)(B(H)))=∞ for all λ∈σπ(△A,B), or, there exists a λ∈σπ(△A,B)∩σp(△A,B); (ii) if X is Hilbert–Schmidt, and AXB−λX=0 for some λ∈σπ(△A,B), then A⁎XB⁎−λ¯X=0; (iii) if V⁎∈B(H) is an isometry, λ∈σπ(A), A−1(0)⊆A⁎−1(0), and AXV−λX=0 (or, AX−λXV=0) for some X∈B(H), then A⁎XV⁎−λ¯X=0 (resp., A⁎X−λ¯XV⁎=0).