Is Fourier series an "inverse" of Taylor series?

Just a brief comparison:

Fourier series are:

• Global in nature. Fourier series are computed using an integral over one period (they represent the entire function over one period even if it is discontinuous, piece-wise continuous, etc.... Gibbs phenomenon not withstanding)
• Fourier series decompose a function by representing it as a linear combination of basis functions (sine and cosine). These basis functions are orthogonal.
• Fourier series are invertable. That is once you have your Fourier coefficients you can reconstruct the entire function from the coefficients (up to a point, i.e. Gibbs phenomenon).

On the other hand:

Taylor series are:

• Local in nature. Taylor series are computed using an infinite number of derivatives at one point (therefore they cannot represent functions which are discontinuous, piece-wise continuous, etc).
• Taylor series decompose a function by representing it as a fixed combination of derivatives. These "basis" functions are not orthogonal.
• Taylor series are invertable only in the neighborhood of a point. You cannot, in general, recover the entire function from a Taylor series.

Your understanding is incorrect in both cases.

A Taylor series is able to represent an "arbitrary" function $f$ in the neigbourhood of a given point $a$ in the domain of $f$ as a power series: $$f(x)=\sum_{k=0}^\infty c_k (x-a)^k\ ,$$ i.e., in the form of an "infinite polynomial". Neither the series nor its finite partial sums are "linear functions". The coefficients $c_k$ of this power series are connected to the represented function $f$ by the formula $c_k=f^{(k)}(a)/k!\ $. So we see here the values $f^{(k)}(a)$ entering in a linear way, but the derivatives $f^{(k)}$ as functions do not appear in the representation.

A Fourier series is able to represent an "arbitrary" periodic function $f$ of period $2\pi$ as an "infinite linear combination" of the basic periodic functions $t\mapsto \sin(k t)$, $t\mapsto \cos(k t)$; so it has the form $$f(t)={a_0\over 2}+\sum_{k=1}^\infty (a_k\cos(kt)+b_k\sin(kt))\ .$$ The coefficients $a_k$, $b_k$ in this representation are connected to the given $f$ via certain integrals (which I won't write down here).

In fact there is a certain connection between these two paradigms. It works in the realm of functions of a complex variable $z$, but there is no question of the same function $f$ of a real variable $x$ or $t$ being represented with more or less the same coefficients $c_k$, $a_k$, $b_k$ first as a Taylor series and then as a Fourier series.

Both Fourier series and Taylor series are decompositions of a function, the difference is that Taylor series are inherently local, while Fourier series are inherently global.

You can find out more here:
Connection between Fourier transform and Taylor series