When the scene root's orientation is set to RIGHT_TO_LEFT, arrow key navigation seems reversed (right arrow moves left and left arrow moves right).

Apparently, this is not a TableView issue, but here is where I encountered it.

Here is my SSCCE:

import javafx.application.Application;
import javafx.beans.property.BooleanProperty;
import javafx.beans.property.SimpleBooleanProperty;
import javafx.beans.property.SimpleStringProperty;
import javafx.beans.property.StringProperty;
import javafx.collections.FXCollections;
import javafx.collections.ObservableList;
import javafx.geometry.NodeOrientation;
import javafx.scene.Scene;
import javafx.scene.control.SelectionMode;
import javafx.scene.control.TableColumn;
import javafx.scene.control.TableView;
import javafx.scene.layout.VBox;
import javafx.stage.Stage;

public class SandBox extends Application {


    @Override
    public void start(Stage stage) throws Exception {
        ObservableList<Person> persons = FXCollections.observableArrayList();
        persons.add(new Person("Sami", "Haddad", false));
        persons.add(new Person("Ahmed", "Hasan", true));
        persons.add(new Person("Rami", "Kassar", true));
        persons.add(new Person("Nehad", "Hamad", false));
        persons.add(new Person("Jamal", "Raei", true));
        persons.add(new Person("Ameer", "Raji", true));
        persons.add(new Person("Tahseen", "Muhsen", true));

        TableView<Person> tableView = new TableView<>();
        tableView.setItems(persons);
        tableView.getSelectionModel().setSelectionMode(SelectionMode.SINGLE);
        tableView.getSelectionModel().setCellSelectionEnabled(true);

        TableColumn<Person, String> colFirstName = new TableColumn<>("First Name");
        colFirstName.setCellValueFactory(p -> p.getValue().firstName);
        colFirstName.setCellFactory(TextFieldTableCell.forTableColumn());
        colFirstName.setOnEditCommit(event -> event.getTableView().getItems().get(event.getTablePosition().getRow()).firstName.set(event.getNewValue()));

        TableColumn<Person, String> colLastName = new TableColumn<>("Last Name");
        colLastName.setCellValueFactory(p -> p.getValue().lastName);
        colLastName.setCellFactory(TextFieldTableCell.forTableColumn());
        colLastName.setOnEditCommit(event -> event.getTableView().getItems().get(event.getTablePosition().getRow()).lastName.set(event.getNewValue()));

        TableColumn<Person, Boolean> colInvited = new TableColumn<>("Invited");
        colInvited.setCellValueFactory(p -> p.getValue().invited);
        colInvited.setCellFactory(javafx.scene.control.cell.CheckBoxTableCell.forTableColumn(colInvited));

        tableView.getColumns().addAll(colFirstName, colLastName, colInvited);
        tableView.setNodeOrientation(NodeOrientation.RIGHT_TO_LEFT);

        stage.setTitle("TableView Selection Bug");
        stage.setScene(new Scene(new VBox(tableView)));
        stage.show();
    }

    class Person {
        public StringProperty firstName;
        public StringProperty lastName;
        public BooleanProperty invited;

        public Person() {
            this.firstName = new SimpleStringProperty("");
            this.lastName = new SimpleStringProperty("");
            this.invited = new SimpleBooleanProperty(false);
        }

        public Person(String fname, String lname, boolean invited) {
            this();
            this.firstName.set(fname);
            this.lastName.set(lname);
            this.invited.set(invited);
        }
    }

    public static void main(String[] args) {
        launch(args);
    }
}

To reproduce the bug:
Select a cell in the table view, then navigate cells using arrow keys.