The three-j-symbol has more symmetric properties that the Clebsch-Gordan coefficient. (The rules for interchange of rows, signs etc are found on p36 in LM.) Three-j symbols also enter the Wigner-Eckart theorem! (2.107 LM, p238-239 Sakurai) The folder for each group will contain one set of tables of 3j-symbols, and you may also want to try out the on-line 3j-calculator at the Weizmann institute in Israel.
The 3j-symbol is represented graphically by three lines, one for each angular momentum, meeting at a vertex. The sign at the vertex corresponds to the order of the angular momenta in the 3j-symbol. (see p48, LM). Additional phase factors are given by arrows on the lines.
The graphical representation for the Clebsch-Gordan coefficient is shown in (3.13) and for the Wigner-Eckart theorem in (3.17). Using these relations, and the rules for handling graphs, gives a convenient way to deals with the angular parts of most problems appearing in atomic physics. Doing the angular momentum this way is fun!
The orthogonality relations (2.86-2.87) between the Clebsch-Gordan coefficients lead to the diagrammatic relations (3.25) and (3.32), which then lead to the "JLV" (Jucys, Levinson and Vanagas) theorems for manipulating graphs.