Inverse DFT to a real signal when only odd harmonics are present
So, I've read the paper I cited more attentively, and it turned out that the author, while not having described the inverse algorithm explicitly, left enough clues in the derivation of the forward algorithm, after all. In case anyone needs it in the future, here it is.

Input: the nonzero (odd) harmonics $X_k$, $k=0,\dots,N/4-1$ (assuming here for simplicity that $N$ is divisible by 4).


Build the array $Y_k$, $k=0,\dots,N/4$: $Y_0 = -2\mathrm{Im}X_0$, $Y_{N/4} = 2\mathrm{Im}X_{N/4-1}$, for the rest $Y_k = i(X_{k+1}-X_k)$.
Calculate $y = \mathrm{IRFFT}(Y)$ (the standard inverse FFT to real signal, e.g. numpy.fft.irfft in Python or dft_c2r from FFTW). The resulting vector $y$ is real and has length $N/2$.
$x_0 = \frac{2}{N}\sum_{k=0}^{N/4-1} \mathrm{Re}X_k$, $x_n = y_n / (4 \sin(2\pi n))$, $n=1,\dots,N/2-1$. This gives the first half of the full vector $x$.


Of course, this method is not ideal, since the division by a sine can introduce instabilities. There may exist a method to deal with that (there is one for the FFT of an (anti)symmetric signal, which requires a similar post-processing).