Effects of interchanging sine terms with cosine terms Suppose we have a real signal $x(t)$. Now, we know that $x(t)$ can be represented as a sum of sines and cosines.
w be the angular frequency. 


If $a(\omega)$ be the coefficients of the cosine terms, and $b(\omega)$ be the coefficients of the sine terms, then if we perform a transformation on this signal such that now $b(\omega)$ are the coefficients of not sines but cosines and similarly, $a(\omega)$ the coefficients of the sines, then how will the waveform transform$[x(t)]$  look like?


What will be the similarity between this transformed waveform and the original waveform?
What is the name for this transformation (is it Hilbert transformation? )



Please attach graphs for an example $x(t)$ to clarify.
If you have a $T$-periodic signal $x(t)$ with a Fourier series expansion

$$x(t)=a_0+\sum_{n=1}^{\infty}\left[a_n\cos(n\omega_0t)+b_n\sin(n\omega_0t)\right],\quad \omega_0=\frac{2\pi}{T}\tag{1}$$

then its Hilbert transform is given by

$$\mathcal{H}\{x(t)\}=\hat{x}(t)=\sum_{n=1}^{\infty}\left[a_n\sin(n\omega_0t)-b_n\cos(n\omega_0t)\right]\tag{2}$$

because $\mathcal{H}\{\cos(t)\}=\sin(t)$, $\mathcal{H}\{\sin(t)\}=-\cos(t)$, and $\mathcal{H}\{c\}=0$ for constant $c$.

This is not exactly the same as just swapping the coefficients $a_n$ and $b_n$. The signal with swapped coefficients (assuming $a_0=0$) is

$$y(t)=\sum_{n=1}^{\infty}\left[a_n\sin(n\omega_0t)+b_n\cos(n\omega_0t)\right]$$

which is related to $\hat{x}(t)$ by

$$y(t)=-\hat{x}(-t)\tag{3}$$

It is also equal to the Hilbert transform of $x(-t)$:

$$y(t)=\mathcal{H}\{x(-t)\}\tag{4}$$

So swapping the coefficients is equivalent to applying a linear time-varying system to the signal. It cannot be achieved by a Hilbert transform alone (which is a time-invariant operation).

Note that for even signals ($b_n=0$) and for odd signals ($a_n=0$) one does indeed obtain the Hilbert transform (for odd signals with a negative sign) by swapping coefficients. This is shown in the example in hops's answer.