When I was experimenting with orthogonalization of polynomials
$$p_n(x)=\begin{cases} 1-x^n&\text{if }n\equiv0\; (\operatorname{mod}2),\\ x-x^n&\text{otherwise}, \end{cases}$$
i.e. simplest binomials vanishing at $x=-1$ and $x=1$, with respect to inner product of
$$\langle p_n,p_m\rangle=\int\limits_{-1}^1 p_n(x)p_m(x)\,dx,$$
i.e. the one used to generate Legendre polynomials, I found that the resulting orthogonal polynomials are (up to normalization) nothing other than associated Legendre polynomials $P_l^m$ with $m=2$.
Similarly, I can get a series of orthogonal polynomials vanishing at the endpoints with dot product the same as that of Chebyshev polynomials. The first several such polynomials are:
$$\begin{align} c_2(x)=&\sqrt{\frac8{3\pi}}(x^2-1),\\ c_3(x)=&\sqrt{\frac{16}\pi}(x^3-x),\\ c_4(x)=&\sqrt{\frac{32}{15\pi}}(6x^4-7x^2+1),\\ c_5(x)=&\sqrt{\frac{16}{3\pi}}(8x^5-11x^3+3x),\\ c_6(x)=&\sqrt{\frac8{35\pi}}(80x^6-128x^4+51x^2-3). \end{align}$$
After some more experiments I've found that these polynomials solve a modified version of Chebyeshev equation, with modification highly resembling that of associated vs usual Legendre polynomials:
$$(1-x^2)y''(x)-xy'(x)+\left(n^2-\frac2{1-x^2}\right)y(x)=0,$$
which, if we omit the $\frac2{1-x^2}$ term, becomes the usual equation for Chebyshev polynomials.
Actually, they can be generated from the Chebyshev polynomials themselves by differentiation (again, inspiration comes from associated Legendre polynomials):
$$c_n(x)=\sqrt{\frac2\pi}\frac{x^2-1}{n\sqrt{n^2-1}}\frac{\mathrm d^2}{\mathrm dx^2}T_n(x).$$
Is this a known series of polynomials? I failed to find anything resembling "associated Chebyshev polynomials", but maybe they have another name?