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In math jargon, perpendicular coordinates are called covariant ones and skew coordinates are called contravariant ones.

["Perpendicular" and "skew" have been my labels. At one point later I've learned that in math jargon, the perpendicular coordinates are called covariant ones or projection ones, and the skew coordinates are called contravariant ones or parallel-axis ones].

Of course, it is possible to plot pattern or structure coordinates while forcing the axes to be geometrically orthogonal on the plot - it is what when you take the table of the loadings (pattern or structure) and give to your software to build a standard scatterplot of those, - but then the angle between the variable vectors will appear widened. And so it will be a distorted loading plot, since the aforesaid original angle was the correlation coefficient between the variables.

In math jargon, perpendicular coordinates are called covariant ones and skew coordinates are called contravariant ones.

Of course, it is possible to plot pattern or structure coordinates while forcing the axes to be geometrically orthogonal on the plot - it is what when you take the table of the loadings (pattern or structure) and give to your software to build a standard scatterplot of those, - but then the angle between the variable vectors will appear widened. And so it will be a distorted loading plot, since the aforesaid original angle was the correlation coefficient between the variables.

Let me recommend you first to read this Q/A. It is about rotations and can hint towards or partly answer your question.

Structure matrix seems to me potentially better than pattern matrix in back interpretation of variables by factors, if such a task arises. And it can rise when we validate items in questionnaire construction, - that is, decide which variables to select and which to drop in the scale being created. Just remember that in psychometry common validity coefficient is correlation (and not regression) coefficient between construct/criterion and item. Usually I include an item in a scale this way: (1) look at maximal correlation (structure matrix) in the item's row; (2) if the value is above a threshold (say, .40), select the item if its situation in pattern matrix confirms the decision (i.e. the item is loaded by the factor - and desirably only by this one - which scale we're constructing). Also factor scores coefficient matrix is what is useful in addition to pattern and structure loadings in the job of selection items for a factor construct.

$^1$ Pattern loadings are the regression coefficients of the factor model equation. It the model, the being predicted variable is meant either standardized (in a FA of correlations) or centered (in a FA of covariances) observed feature, while the factors are meant standardized (with variance 1) latent features. Coefficients of that linear combination are the pattern matrix values. As comes clear from pictures below here - pattern coefficients are never greater than structure coefficients which are correlations or covariances between the being predicted variable and the standardized factors.

Some geometry. Loadings are coordinates of variables (as their vector endpoints) in the factor space. We use to encounter those on "loading plots" and "biplots". See formulas.

See detailed explanation of a loading plot (in settings of orthogonal factors) here.

Let me recommend you first to read this Q/A. It is about rotations and can hint towards or partly answer your question.

Structure matrix seems to me potentially better than pattern matrix in back interpretation of variables by factors, if such a task arises. And it can rise when we validate items in questionnaire construction, - that is, decide which variables to select and which to drop in the scale being created. Just remember that in psychometry common validity coefficient is correlation (and not regression) coefficient between construct/criterion and item. Usually I include an item in a scale this way: (1) look at maximal correlation (structure matrix) in the item's row; (2) if the value is above a threshold (say, .40), select the item if its situation in pattern matrix confirms the decision (i.e. the item is loaded by the factor - and desirably only by this one - which scale we're constructing). Also factor scores coefficient matrix is what is useful in addition to pattern and structure loadings in the job of selection items for a factor construct.

$^1$ Pattern loadings are the regression coefficients of the factor model equation. It the model, the being predicted variable is meant either standardized (in a FA of correlations) or centered (in a FA of covariances) observed feature, while the factors are meant standardized (with variance 1) latent features. Coefficients of that linear combination are the pattern matrix values. As comes clear from pictures below here - pattern coefficients are never greater than structure coefficients which are correlations or covariances between the being predicted variable and the standardized factors.

Some geometry. Loadings are coordinates of variables (as their vector endpoints) in the factor space. We use to encounter those on "loading plots" and "biplots". See formulas.

See detailed explanation of a loading plot (in settings of orthogonal factors) here.